Freefall in curved space
2
Contents:
1.
Curvilinear coordinate systems .....................................................................................4
1.1.
1.2.
Non orthogonal example ....................................................................................................... 5
General curvilinear systems ................................................................................................ 9
2.
Tensors................................................................................................................................. 13
3.
Christoffel symbols, 1st and 2nd kind .......................................................................... 16
3.1.
Christoffel symbols as Derivatives of the Metric Tensor ........................................ 17
4.
Covariant derivative ........................................................................................................ 19
5.
The geodesic equation .................................................................................................... 20
6.
Appendix A, a wakeup problem................................................................................... 27
7.
Appendix B, index notation........................................................................................... 31
8.
Appendix C, remarks of interest ................................................................................. 36
9.
Appendix D, T-shirt from CERN ................................................................................... 39
10.
References ........................................................................................................................ 41
3
Abstract
This article derives the equation for a geodesic in a curved
space. It assumes that the reader has a working knowledge of
partial derivatives, the chain rule and some basics about vector
algebra/calculus and variational calculus. It tries to introduce
the concepts and notations that are necessary to be able to
read and understand the different terms of the equation. Here is
the ‘layout’ of the article:
- short about curvilinear coordinate systems
- some facts about tensors and why tensors are of
fundamental importance in physics
- Christoffel symbols
- covariant derivative of a covariant and a contravariant
vector
- derivation of the equation for a geodesic in curved space
As an introduction the reader is invited to look at a simple
example in Appendix A and if unfamiliar with index notation also
read Appendix B. Appendix C and D contains miscellaneous
information about mathematically related areas.
4
1. Curvilinear coordinate
systems
A useful observation :
𝜙 = 𝜙(𝑥1 , 𝑥2 , 𝑥3 ) is a scalar function in 3 dimensions. For
each point in space it has a scalar value. Setting 𝜙(𝑥1 , 𝑥2 , 𝑥3 )
= const.= c1 will introduce a dependency between the
variables 𝑥1 , 𝑥2 and 𝑥3 and thereby define a surface in the 3d
space. A point on the surface is represented by a vector 𝑟⃗ . A
small variation of 𝑥1 , 𝑥2 and 𝑥3 under the constraint
𝜙(𝑥1 , 𝑥2 , 𝑥3 ) = c1 will give a new point on the surface
represented by the vector 𝑟⃗ + d𝑟⃗. The vector d𝑟⃗ will clearly
“lie in surface”. The situation is shown in the figure below.
x3
dr
r
x2
x1
d(c1) = 0 = 𝑑𝜙 =
𝜕𝜙
𝜕𝑥𝑖
𝑑𝑥𝑖 = ∇ϕ ∙ 𝑑𝑟⃗
=>
the vector ∇ϕ is ⊥ to the surface 𝜙(𝑥1 , 𝑥2 , 𝑥3 ) = c1
5
An example concerning dependence:
In 3 dimensional space where the cross product of vectors is
defined, two functions 𝑢 = 𝑢(𝑥1 , 𝑥2 , 𝑥3 ) and 𝑣 =
𝑣(𝑥1 , 𝑥2 , 𝑥3 ) are not independent.
There exists a function 𝑓 = 𝑓(𝑢, 𝑣) = const. = c1 that
connects the two. A necessary and sufficient condition is that
∇𝑣 × ∇𝑢 = 0.
0 = ∇𝑓 = 𝑢∇𝑣 + 𝑣∇𝑢 => ∇𝑣 and ∇𝑢 are parallel => ∇𝑣
∇𝑢 = 0 . This last expression is the condition for the
determinant of a Jacobian matrix to be zero. This
interdependence condition is also used in the Legendre
transform.
1.1. Non orthogonal example
We start with a simple example of a non orthogonal
coordinate system.
𝑥2
𝑞2 = 𝑥2 − 𝑥1
r
𝑥1
𝑞1 = 𝑥1
×
6
The coordinate transformation is given by
𝑥1 = 𝑞1
𝑥2 = 𝑞1 + 𝑞2
And its inverse
𝑞1 = 𝑥1
𝑞2 = 𝑥2 − 𝑥1
There are two “natural” ways to specify the coordinate
components for the vector 𝑟⃗ = (𝑥1 , 𝑥2 ) in the oblique
coordinate system.
Contravariant components =>
r
The basis vectors are
⃗⃗
𝜕𝒓
𝜕𝑞1
= (1,1) and
⃗⃗
𝜕𝒓
𝜕𝑞2
Call these covariant basis vectors 𝜀⃗𝑖 =
and the
𝑟⃗ vector can be written as
= (0,1)
⃗⃗
𝜕𝒓
𝜕𝑞𝑖
i = 1..2
7
𝑟⃗ = 𝑞𝑖
⃗⃗
𝜕𝒓
𝜕𝑞𝑖
= 𝑞𝑖 𝜀⃗𝑖 i=1..2 , where the 𝑞𝑖
are the vectors
contravariant components.
Covariant components =>
90°
r
90°
The basis vectors are ∇𝑞1
= (1,0) and ∇𝑞2 = (−1,1)
Call these contravariant basis vectors 𝜀⃗𝑖 = ∇𝑞𝑖 i = 1..2
and the 𝑟⃗ vector can be written as
𝑟⃗ = 𝑞𝑖 ∇𝑞𝑖 = 𝑞𝑖 𝜀⃗𝑖 i=1..2 , where the 𝑞𝑖 are the vectors
covariant components.
Notice now the joyful fact that
[ ∇𝑞𝑖 ∙
⃗⃗
𝜕𝒓
𝜕𝑞𝑗
=
𝜕𝑞𝑖 𝜕𝑥𝑘
𝜕𝑥𝑘 𝜕𝑞𝑗
𝜀⃗𝑖 ∙ 𝜀⃗𝑗 = ∇𝑞𝑖 ∙
= 𝑐ℎ𝑎𝑖𝑛 𝑟𝑢𝑙𝑒 =
𝜕𝑞𝑖
𝜕𝑞𝑗
⃗⃗
𝜕𝒓
𝜕𝑞𝑗
= 𝛿𝑗𝑖
= 𝛿𝑗𝑖 ]
The two sets of basis vectors are called reciprocal.
Using the two reciprocal sets of basis vectors, the scalar
product of two vectors simplifies to
8
𝑎⃗ ∙ 𝑏⃗⃗ = 𝑎𝑖 𝜀⃗𝑖 ∙ 𝑏 𝑘 𝜀⃗𝑘 = 𝑎𝑖 𝑏 𝑘 𝜀⃗𝑖 ∙ 𝜀⃗𝑘 = 𝑎𝑖 𝑏 𝑘 𝛿𝑘𝑖 = 𝑎𝑖 𝑏 𝑖
Notice that this is now valid in a general curvilinear coordinate
system. In Cartesian coordinates the two sets of vector
components are equal.
This simple example should clarify why tensor calculus and
general curvilinear coordinate systems needs the covariant and
contravariant concept(s).
Concepts that are linked to reciprocal basis vectors are dual
basis vectors (http://en.wikipedia.org/wiki/Dual_basis) and dual
vector spaces (http://en.wikipedia.org/wiki/Dual_space ).
9
1.2. General curvilinear systems
Generalized coordinates are often denoted by a q and a sub
index, like this, 𝑞𝑖 (), and are a coordinate transformation from a
Cartesian orthogonal coordinate system 𝑥𝑖 :
𝑥𝑖 = 𝑥𝑖 ( 𝑞1 , 𝑞2 , …, 𝑞𝑁 ) i = 1..N
[1-1]
The functions 𝑥𝑖 must be differentiable and form an
independent set, meaning that the Jacobian determinant must
be # 0. The Jacobian is defined as the matrix (Jacobian often
means Jacobian determinant):
𝜕𝑥1
𝜕𝑞1
⋮
𝜕𝑥𝑁
…
⋱
…
𝜕𝑥1
𝜕𝑞𝑁
⋮
and is often written as
𝜕𝑥𝑁
[ 𝜕𝑞1
𝜕𝑞𝑁 ]
𝜕(𝑥1 ,…𝑥)
𝐽𝑞 (𝑥1 , … 𝑥𝑁 ) ,
𝜕(𝑞1 ,…𝑞𝑁 )
or/and some more versions.
and the corresponding determinants with bars around in the
𝜕𝑥𝑖
usual way or det(𝜕𝑞
𝑗
). If the Jacobian determinant is # 0 the
functions 𝑥𝑖 can be expressed in 𝑞1 , … , 𝑞𝑛 :
𝑞𝑖 = 𝑞𝑖 ( 𝑥1 , 𝑥2 , …, 𝑥𝑁 ) i = 1..N
[1-2]
A line element, displacement vector, in curvilinear
coordinates is given by (chain rule):
𝑑𝑟⃗ =
𝜕𝑟⃗
𝜕𝑞𝑖
𝑑𝑞𝑖
[1-3]
10
The vectors
⃗⃗
𝜕𝒓
𝜕𝑞𝑖
provide a natural vector basis and can be
visualized starting with a vector 𝑟⃗ and varying one 𝑞𝑖 at a time.
𝜕𝑟⃗
𝑟⃗
𝜕𝑞1
, 𝑞2 𝑎𝑛𝑑 𝑞3 𝑓𝑖𝑥𝑒𝑑
The reciprocal vector basis is ∇𝑞𝑖 , the ∇𝑞𝑖 vectors, can be
visualized by having 𝑞𝑖 fixed and vary the other two. The vector
∇𝑞𝑖 is perpendicular to the surface generated in this way.
vector ∇𝑞1
dr Surface: 𝑞1 𝑓𝑖𝑥𝑒𝑑
𝑟⃗
Denote
⃗⃗
𝜕𝒓
𝜕𝑞𝑖
= 𝜀⃗𝑖
and ∇𝑞𝑖 = 𝜀⃗𝑖 and that they are reciprocal
basis vectors can be easily shown:
11
𝜀⃗𝑖 ∙ 𝜀⃗𝑗 = ∇𝑞𝑖 ∙
𝜕𝑞𝑖
𝜕𝑞𝑗
⃗⃗
𝜕𝒓
=
𝜕𝑞𝑗
𝜕𝑞𝑖 𝜕𝑥𝑘
𝜕𝑥𝑘 𝜕𝑞𝑗
= 𝑐ℎ𝑎𝑖𝑛 𝑟𝑢𝑙𝑒 =
= 𝛿𝑗𝑖
The
⃗⃗
𝜕𝒓
𝜕𝑞𝑖
[1-4]
= 𝜀⃗𝑖
basis is particularly appropriate for vectors such
as the velocity. The velocity components
coordinates are simply
𝑑𝑟⃗
𝑑𝑡
= 𝑥𝑖̇ 𝑥̂𝑖 = 𝑞̇ 𝑘
𝑞𝑖̇ in the
⃗⃗
𝜕𝒓
𝜕𝑞𝑖
𝑥𝑖̇
in Cartesian
system :
⃗⃗
𝜕𝒓
𝜕𝑞𝑘
On the other hand, the ∇𝑞𝑖 = 𝜀⃗𝑖 basis is appropriate for the
gradient operator as, by the chain rule again, the components
of the gradient are simply
∇𝑓 =
𝜕𝑓
𝜕𝑥𝑖
𝑥̂𝑖 =
𝜕𝑓
𝜕𝑞𝑘
𝜕𝑓
𝜕𝑞𝑖
in the ∇𝑞𝑘 basis :
∇𝑞𝑘
In general, any vector can be expanded in terms of either
basis.
𝑣⃗ = 𝑣𝑖 𝜀⃗𝑖 = 𝑣 𝑖 𝜀⃗𝑖
The vector components with a sub index are called the
covariant components and the ones with a super index are
called the contravariant components.
The scalar products
⃗⃗ 𝜕𝒓
⃗⃗
𝜕𝒓
𝜕𝑞𝑖 𝜕𝑞𝑗
= 𝜀⃗𝑖 𝜀⃗𝑗 form a second-rank
tensor that describes all the angles between the basis vectors
and all the lengths of the vectors. It is called the covariant
metric tensor and is denoted by 𝑔𝑖𝑗 . It is obviously symmetric.
𝑔𝑖𝑗 = 𝑔𝑗𝑖 =
⃗⃗ 𝜕𝒓
⃗⃗
𝜕𝒓
𝜕𝑞𝑖 𝜕𝑞𝑗
= 𝜀⃗𝑖 𝜀⃗𝑗
[1-5]
The scalar products ∇𝑞𝑖 ∇𝑞𝑗 = 𝜀⃗𝑖 𝜀⃗𝑗 form a second-rank
tensor that describes all the angles between the basis vectors
12
and all the lengths of the vectors. It is called the contravariant
metric tensor and is denoted by 𝑔𝑖𝑗 . It is obviously
symmetric.
𝑔𝑖𝑗 = 𝑔 𝑗𝑖 = ∇𝑞𝑖 ∇𝑞𝑗 = 𝜀⃗𝑖 𝜀⃗𝑗
[1-6]
The concept of metric tensor is one of the cornerstones of
geometry in general curvilinear coordinate systems and
differential geometry. The value of the metric tensor is a
function of the position in space, i.e. it’s values are generally
different at different positions, 𝑔𝑖𝑗 = 𝑔𝑖𝑗 (𝑥1 , … , 𝑥𝑁 ) and 𝑔𝑖𝑗 =
𝑔𝑖𝑗 (𝑥1 , … , 𝑥𝑁 )
One frequent use is that it can lower and rise the indices of the
components of a vector.
𝑣⃗ = 𝑣𝑘 𝜀⃗𝑘 = 𝑣 𝑗 𝜀⃗𝑗 => 𝑣𝑘 𝜀⃗𝑘 ∙ 𝜀⃗𝑖 = 𝑣 𝑗 𝜀⃗𝑗 ∙ 𝜀⃗𝑖 =>
[𝜀⃗𝑘 ∙ 𝜀⃗𝑖 = 𝛿𝑖𝑘 ] => 𝑣𝑖 = 𝑔𝑖𝑗 𝑣 𝑗
𝑣𝑖 = 𝑔𝑖𝑗 𝑣 𝑗 and similarly 𝑣 𝑖 = 𝑔𝑖𝑗 𝑣𝑗
[1-7]
Example:
𝑑𝑠 2 = 𝑑𝑠⃗ ∙ 𝑑𝑠⃗ = 𝑑𝑞𝑖 𝑑𝑞𝑖 = 𝑔𝑖𝑗 𝑑𝑞𝑖 𝑑𝑞 𝑗
[1-8]
Example:
𝑔𝑘𝑙 𝑔𝑖𝑘 = 𝑔𝑖𝑘 𝜀⃗𝑘 ∙ 𝜀⃗𝑙 = 𝜀⃗𝑖 ∙ 𝜀⃗𝑙 = 𝛿𝑖𝑙
[1-9]
13
2.
Tensors
This chapter notes some facts about tensors that are more or
less relevant to this article. The important points are marked
with a @ symbol.
Einstein’s summation convention is used.
A tensor can be seen as a generalization of the vector
concept and is defined by how it is transformed by a coordinate
transformation.
The definition of tensors via transformation properties
conforms to the physicist’s notion that physical observables
must not depend on the choice of coordinate frames.
@ The “main theorem” of tensor calculus is as follows:
If two tensors of the same type are equal in one coordinate
system, then they are equal in all coordinate systems.
@ A tensor has a rank.
A scalar has rank 0 and is an “invariant object in the space
with respect to the group of coordinate transformations”.
A vector is a rank 1 tensor with covariant components 𝑉𝑖 and
contravariant components 𝑉 𝑖 .
Examples of rank 2 tensors are the metric tensor 𝑔𝑖𝑗
(covariant) 𝑔𝑖𝑗 (contravariant), the inertia tensor 𝐼𝑖𝑗
𝑖
and 𝛿𝑗 a rank 2 mixed tensor
(http://mathworld.wolfram.com/KroneckerDelta.html)
@ Definition of a rank 1 tensor:
Taking a differential distance vector 𝑑𝑟⃗ and letting
function of the unprimed variables.
𝑖
𝑑𝑥′ =
𝜕𝑥′𝑖
𝜕𝑥 𝑘
𝑑𝑥 𝑘
Any set of quantities 𝐴 𝑗 that transform according to
𝑑𝑥′𝑖
be a
[2-1]
14
𝜕𝑥′𝑖
𝑖
𝐴′ =
𝜕𝑥
𝑗
𝐴
𝑗
[2-2]
is defined as a contravariant vector (contravariant rank-1
tensor), and the indices are written as superscript.
Taking a scalar field 𝜙 the transformation is different.
∇𝜙 =
𝜕𝜙′
𝜕𝑥′𝑖
=
𝜕𝜙
𝜕𝑥 𝑖
𝑥̂ 𝑖 =>
𝜕𝜙 𝜕𝑥 𝑗
[2-3]
𝜕𝑥 𝑗 𝜕𝑥′𝑖
Notice the difference, it is vital. [2-3] is the definition of a
covariant vector. Rewritten it looks like this
𝐴′𝑖 =
𝜕𝑥 𝑗
𝜕𝑥′𝑖
𝐴𝑗
[2-4]
In the same way tensor rank 2 is defined. When the rank is 2
there is also mixed tensors, one subscript index and one
superscript.
@ Definition of a rank 2 tensor:
𝑖𝑗
𝐴′ =
𝐵′𝑗𝑖 =
𝜕𝑥′𝑖 𝜕𝑥′𝑗
𝜕𝑥 𝑘 𝜕𝑥 𝑙
𝜕𝑥′𝑖 𝜕𝑥 𝑙
𝜕𝑥 𝑘
𝐶′𝑖𝑗 =
𝜕𝑥′𝑗
𝐴𝑘𝑙
𝐵𝑙𝑘
𝜕𝑥 𝑘 𝜕𝑥 𝑙
𝜕𝑥′𝑖 𝜕𝑥′𝑗
𝐶𝑘𝑙
[2-5]
[2-6]
[2-7]
@ The quotient rule.
To establish the tensor nature of a quantity can be tedious.
Help comes from the quotient rule:
If A and B are tensors, and if the expression A = BT is invariant
15
under coordinate transformation, then T is a tensor.
Example: If A and B are tensors and the expression holds in all
(rotated) Cartesian coordinate systems, then K is a tensor in the
following expressions.
𝐾𝑖 𝐴𝑖 = 𝐵 and 𝐾𝑖𝑗 𝐴𝑗 = 𝐵𝑖 and 𝐾𝑖𝑗 𝐴𝑗𝑘 = 𝐵𝑖𝑘 and
𝐾𝑖𝑗𝑘𝑙 𝐴𝑖𝑗 = 𝐵𝑘𝑙
and
𝐾𝑖𝑗 𝐴𝑘 = 𝐵𝑖𝑗𝑘
@ Contraction
Dealing with vectors in orthogonal coordinates the scalar
product is
𝑎⃗ ∙ 𝑏⃗⃗ = (𝑎𝑖 𝑒⃗𝑖 ) ∙ (𝑏𝑗 𝑒⃗𝑗 ) = 𝑎𝑖 𝑏𝑗 𝑒⃗𝑖 ∙ 𝑒⃗𝑗 = 𝑎𝑖 𝑏𝑗 𝛿𝑖𝑗 =
𝑎𝑖 𝑏𝑖 ,with an implicit summation over i
The generalization of this in tensor analysis is a process known
as contraction. Two indices, one covariant and the other
contravariant, are set equal to each other, and then (as implied
by the summation convention) we sum over this repeated index.
The scalar product in a general coordinate system:
𝑗
𝑎⃗ ∙ 𝑏⃗⃗ = 𝑎𝑖 𝜀⃗𝑖 ∙ 𝑏 𝑗 𝜀⃗𝑗 = 𝑎𝑖 𝑏 𝑗 𝜀⃗𝑖 ∙ 𝜀⃗𝑗 = 𝑎𝑖 𝑏 𝑗 𝛿𝑖 =
𝑎𝑖 𝑏 𝑖 = 𝑎𝑖 𝑏𝑖
16
3. Christoffel symbols, 1st and
2nd kind
Normal partial derivatives of a vector doesn’t transform as
tensors under general curvilinear coordinate transformations.
An important property of tensors is that if two tensors A and B
are equal, A=B, in one coordinate system then the transformed
tensors are equal, A’ = B’.
This property means that two different observers in different
coordinate systems agree on physical laws. Substituting regular
partial derivatives with the covariant derivatives, which follows
tensor transformation rules, is therefore important and has been
stated as the mathematical statement of Einstein’s equivalence
principle.
The covariant derivative is defined in the next chapter.
The Christoffel symbols have to be defined first.
Starting with scalar ∅
𝑑∅ =
𝜕∅ 𝑖
𝑑𝑞
𝜕𝑞 𝑖
Since the 𝑑𝑞 𝑖 are the components of a contravariant vector, the
partial derivatives must form a covariant vector by the quotient
rule. The gradient of the scalar becomes
∇∅ =
𝜕∅
𝜕𝑞𝑖
𝜀⃗𝑖
[3-1]
Moving on to the derivatives of a vector, the situation is more
complicated because the basis vectors 𝜀⃗𝑖 and 𝜀⃗𝑖 are not
constant. With vector ⃗⃗⃗⃗
𝑉′ = 𝑉 𝑚 𝜀⃗𝑚 , 𝑉′𝑖 =
⃗⃗ ′
𝜕𝑉
𝜕𝑞𝑗
=
𝜕𝑉 𝑖
⃗⃗𝑖
𝜕𝜀
𝜕𝑞
𝜕𝑞𝐽
𝑖
𝜀
⃗
+
𝑉
𝑖
𝑗
𝜕𝑥 𝑖
𝜕𝑞 𝑚
𝑉 𝑚 , we get
[3-2]
or in component form, direct differentiation
𝜕𝑉′𝑘
𝜕𝑞𝑗
=
𝜕𝑥 𝑘 𝜕𝑉 𝑖
𝜕𝑞𝑖
𝜕𝑞 𝑗
+
𝜕2 𝑥 𝑘
𝜕𝑞𝑗 𝜕𝑞𝑖
𝑉𝑖
[3-3]
17
The right hand side of [3-3] differs from the transformation law
for a second-rank mixed tensor by the second term containing
second derivatives of the coordinates 𝑥 𝑘 .
⃗⃗𝑖
𝜕𝜀
will be some linear combination of
𝜕𝑞𝐽
⃗⃗𝑖
𝜕𝜀
𝜕𝑞𝐽
𝜀⃗𝑘 , write this as
= Γ𝑖𝑗𝑘 𝜀⃗𝑘
[3-4]
Multiply by 𝜀⃗𝑚 and use 𝜀⃗𝑘
Γ𝑖𝑗𝑚 = 𝜀⃗𝑚 ∙
∙ 𝜀⃗𝑚 = 𝛿𝑘𝑚
to get
⃗⃗𝑖
𝜕𝜀
[3-5]
𝜕𝑞𝐽
These are Christoffel symbols of the second kind. They
are not third-rank tensors. And
𝜕𝑉 𝑖
𝜕𝑞𝑗
is not generally a second-
rank tensor.
⃗⃗𝑖
𝜕𝜀
𝜕𝑞𝑗
=
𝜕2 𝑟⃗
𝜕𝑞𝑗 𝜕𝑞𝑖
=
𝜕2 𝑟⃗
𝜕𝑞𝑖 𝜕𝑞𝑗
=
⃗⃗𝑗
𝜕𝜀
𝜕𝑞𝑖
, meaning that these
Christoffel symbols are symmetric in the lower indices.
Γ𝑖𝑗𝑚 = Γ𝑗𝑖𝑚
[3-6]
Christoffel symbols of the first kind can be defined as
[𝑖𝑗, 𝑘] = 𝑔𝑚𝑘 Γ𝑖𝑗𝑚
[3-7]
The symmetry [ij, k] = [ji, k] follows from second kinds
symmetry. [ij, k] is not a third-rank tensor.
[𝑖𝑗, 𝑘] = 𝑔𝑚𝑘 𝜀⃗𝑚 ∙
⃗⃗𝑖
𝜕𝜀
𝜕𝑞𝐽
= [𝜀⃗𝑘 = 𝑔𝑚𝑘 𝜀⃗𝑚 ] = 𝜀⃗𝑘 ∙
⃗⃗𝑖
𝜕𝜀
𝜕𝑞𝐽
… [3-8]
.
3.1. Christoffel symbols as Derivatives of
the Metric Tensor
𝑔𝑖𝑗 = 𝜀⃗𝑖 ∙ 𝜀⃗𝑗
[ definition of covariant metric tensor ]
18
Differentiate to get:
𝜕𝑔𝑖𝑗
𝜕𝑞𝑘
=
⃗⃗𝑖
𝜕𝜀
𝜕𝑞𝑘
∙ 𝜀⃗𝑗 + 𝜀⃗𝑖 ∙
⃗⃗𝑗
𝜕𝜀
= [equation [3-8]] = [ik, j] +
𝜕𝑞𝑘
[jk, i]
[3-9]
Equation [3-9] yields
1 𝜕𝑔𝑖𝑘
2 𝜕𝑞 𝑗
[ij, k] = {
+
𝜕𝑔𝑗𝑘
𝜕𝑞𝑖
−
𝜕𝑔𝑖𝑗
𝜕𝑞 𝑘
}
[3-10]
This is the sought expression for Christoffel symbols of the first
kind.
Using equation [3-7] :
𝑠 𝑚
𝑔𝑘𝑠 [𝑖𝑗, 𝑘] = 𝑔𝑘𝑠 𝑔𝑚𝑘 Γ𝑖𝑗𝑚 = 𝛿𝑚
Γ𝑖𝑗 = Γ𝑖𝑗𝑠
[3-11]
Equations [3-10] and [3-11] gives the sought expression for
Christoffel symbols of the second kind.
Γ𝑖𝑗𝑠
=
1
𝜕𝑔
𝑔𝑘𝑠 { 𝑖𝑘𝑗
2
𝜕𝑞
+
𝜕𝑔𝑗𝑘
𝜕𝑞𝑖
−
𝜕𝑔𝑖𝑗
𝜕𝑞𝑘
}
[3-12]
19
4.
Covariant derivative
Equation [3-2] :
⃗⃗ ′
𝜕𝑉
𝜕𝑞𝑗
=
𝜕𝑉 𝑖
⃗⃗𝑖
𝜕𝜀
𝜕𝑞
𝜕𝑞𝐽
𝑖
𝜀
⃗
+
𝑉
𝑖
𝑗
can now be
rewritten using the Christoffel symbols
⃗⃗ ′
𝜕𝑉
𝜕𝑞𝑗
=
𝜕𝑉 𝑖
𝜕𝑞𝑗
𝜀⃗𝑖 + 𝑉 𝑖 Γ𝑖𝑗𝑘 𝜀⃗𝑘
and in the last term the k and i
indices are dummy indices, change k -> i and i -> k to get
⃗⃗ ′
𝜕𝑉
𝜕𝑞𝑗
=
𝜕𝑉 𝑖
𝜕𝑞𝑗
𝜀⃗𝑖 + 𝑉
𝑘
𝑖
Γ𝑘𝑗
𝜀⃗𝑖 = (
𝜕𝑉 𝑖
𝜕𝑞𝑗
𝑖
+ 𝑉 𝑘 Γ𝑘𝑗
) 𝜀⃗𝑖
[4-1]
The expression within the parentheses is the covariant
derivative of the contravariant vector 𝑉 𝑖 and the notation for
derivation is a semicolon, not a comma as in chapter about
index notation.
𝑉;𝑗𝑖
=
𝜕𝑉 𝑖
𝜕𝑞𝑗
𝑖
+ 𝑉 𝑘 Γ𝑘𝑗
[4-2]
𝑉;𝑗𝑖 is the covariant derivative of a contravariant vector. It is a
second-rank tensor.
𝑖
By differentiation of the relation 𝜀⃗𝑖 ∙ 𝜀⃗𝑗 = 𝛿𝑗 it is quite easy to
get the expression for the covariant derivative of a covariant
vector.
𝑉𝑖;𝑗 =
𝜕𝑉𝑖
𝜕𝑞𝑗
− 𝑉𝑘 Γ𝑖𝑗𝑘
[4-3]
𝑉𝑖;𝑗
is the covariant derivative of a covariant vector. It is a
second-rank tensor.
⃗⃗⃗⃗ becomes
A differential 𝑑𝑉′
⃗⃗⃗⃗ =
𝑑𝑉′
⃗⃗⃗⃗⃗
𝜕𝑉′
𝜕𝑞𝑗
𝑑𝑞 𝑗 = [ 𝑉;𝑗𝑖 𝑑𝑞 𝑗 ]𝜀⃗𝑖
[4-4]
In Cartesian coordinates the Christoffel symbols vanish and
the ordinary partial derivative coincide with the covariant
derivative.
20
A more detailed proof that the covariant derivative is a tensor
can be found in [Heinbockel] .
Rules for covariant differentiation:
- The covariant derivative of a sum is the sum of covariant
derivatives
- The covariant derivative of a product of tensors is the first
times the covariant derivative of the second plus the
second times the covariant derivative of the first.
- Higher derivatives are defined as derivatives of derivatives.
But take care, in general 𝐴𝑖;𝑗𝑘 ≠ 𝐴𝑖;𝑘𝑗 .
5.
The geodesic equation
A geodesic in Euclidean space is a straight line. In general, it
is the curve of shortest length between two points and the curve
along which a freely falling particle moves. The ellipses of
planets are geodesics around the sun, and the moon is in free
fall around the Earth on a geodesic. The geodesic can be
obtained in a number of ways. We show three of them.
#1 The geodesic can be obtained from variational principles,
[Arfken] 6th Edition.
𝛿 ∫ 𝑑𝑠 = 0
[5-1]
where 𝑑𝑠 2 is the metric of the space.
2
(𝑑𝑠)2 = 𝑑𝑟⃗ ∙ 𝑑𝑟⃗ = (𝜀⃗𝑖 𝑑𝑞𝑖 ) = 𝜀⃗𝑖 ∙ 𝜀⃗𝑗 𝑑𝑞𝑖 𝑑𝑞 𝑗
= 𝑔𝑖𝑗 𝑑𝑞𝑖 𝑑𝑞 𝑗
The variation of 𝑑𝑠 2
𝛿(𝑑𝑠 2 ) = 2𝑑𝑠𝛿𝑑𝑠 = 𝛿(𝑔𝑖𝑗 𝑑𝑞𝑖 𝑑𝑞 𝑗 ) = [𝑑 (𝑢𝑣𝑤) =
(𝑣𝑤)𝑑𝑢 + (𝑢𝑤)𝑑𝑣 + (𝑢𝑣 )𝑑𝑤] = 𝑑𝑞𝑖 𝑑𝑞 𝑗 𝛿𝑔𝑖𝑗 +
𝑔𝑖𝑗 𝑞𝑖 𝛿𝑞 𝑗 + 𝑔𝑖𝑗 𝑞 𝑗 𝛿𝑞𝑖
[5-2]
21
Inserting [5-2] into [5-1] yields
1
∫[
2
𝑔𝑖𝑗
𝑑𝑞𝑖 𝑑𝑞𝑗
𝑑𝑠 𝑑𝑠
𝑑𝑞𝑗 𝑑
𝑑𝑠 𝑑𝑠
𝛿𝑔𝑖𝑗 + 𝑔𝑖𝑗
𝑑𝑞𝑖 𝑑
𝑑𝑠 𝑑𝑠
𝛿𝑑𝑞 𝑗 +
𝛿𝑑𝑞𝑖 ]𝑑𝑠 = 0.
[5-3]
where ds measures the length on the geodesic.
The variations 𝛿𝑔𝑖𝑗 expressed in terms of the independent
variations 𝛿𝑑𝑞 𝑘 yields
𝛿𝑔𝑖𝑗 =
𝜕𝑔𝑖𝑗
𝜕𝑞𝑘
𝛿𝑑𝑞𝑘 = (𝜕𝑘 𝑔𝑖𝑗 )𝛿𝑑𝑞𝑘
[5-4]
Insert [5-4] in equation [5-3], shift the derivatives in the last
two terms of [5-3] upon integrating by parts and rename the
dummy summation indices and [5-3] will be turned into
1
∫[
2
0
𝑑𝑞𝑖 𝑑𝑞𝑗
𝑑𝑠 𝑑𝑠
𝜕𝑘 𝑔𝑖𝑗 −
𝑑
𝑑𝑠
(𝑔𝑖𝑘
𝑑𝑞𝑖
𝑑𝑠
+ 𝑔𝑘𝑗
𝑑𝑞𝑗
𝑑𝑠
) ] 𝛿𝑑𝑞𝑘 𝑑𝑠 =
… [5-5]
The 𝛿𝑑𝑞 𝑘 can have any value, which means that the
integrand of [5-5], set equal to zero, gives the geodesic
equation. It needs some more manipulations, though …
𝑑𝑞𝑖 𝑑𝑞𝑗
𝑑𝑠 𝑑𝑠
𝜕𝑘 𝑔𝑖𝑗 −
𝑑𝑔𝑖𝑘
𝑑
𝑑𝑠
(𝑔𝑖𝑘
𝑑𝑞 𝑗
𝑑𝑞 𝑖
𝑑𝑠
+ 𝑔𝑘𝑗
𝑑𝑞𝑗
𝑑𝑠
)=0
𝑑𝑔𝑘𝑗
= (𝜕𝑗 𝑔𝑖𝑘 ) 𝑑𝑠 and 𝑑𝑠 = (𝜕𝑖 𝑔𝑘𝑗 )
and that 𝑔𝑖𝑗 is symmetric results in :
Using
𝑑𝑠
1 𝑑𝑞𝑖 𝑑𝑞𝑗
2 𝑑𝑠 𝑑𝑠
(𝜕𝑘 𝑔𝑖𝑗 − 𝜕𝑗 𝑔𝑖𝑘 − 𝜕𝑖 𝑔𝑘𝑗 ) − 𝑔𝑖𝑘
𝑑𝑞 𝑖
[5-6]
𝑑𝑠
𝑑2𝑞𝑖
𝑑𝑠 2
in [5-6]
=0
.
… [5-7]
Multiplying [5-7] with 𝑔𝑘𝑙 and using the fact that 𝑔𝑖𝑘 𝑔𝑘𝑗 = 𝛿𝑗𝑖
finally yields the geodesic equation:
𝑑2 𝑞𝑙
𝑑𝑠 2
+
𝑑𝑞𝑖 𝑑𝑞 𝑗 1
𝑑𝑠 𝑑𝑠 2
𝑔𝑘𝑙 (𝜕𝑖 𝑔𝑘𝑗 + 𝜕𝑗 𝑔𝑖𝑘 − 𝜕𝑘 𝑔𝑖𝑗 ) = 0
… [5-8]
𝑙
The coefficient of the velocities is the Christoffel symbol Γ𝑖𝑗
22
#2 An alternative derivation of the geodesic equation can be
found in [Arfken] 7th Edition.
The distance between two points can be represented as
𝐽 = ∫ √𝑔𝑖𝑗 𝑞̇ 𝑖 𝑞̇ 𝑗 𝑑𝑢
[5-9]
To find the geodesic equation it is possible to start from the
action:
𝑆 = ∫ 𝐿 𝑑𝑡
[5-10]
Using the Lagrangian formulation of relativistic mechanics
where, for a particle not subject to a potential other than a
gravitational force (which is described by the metric tensor), the
Lagrangian reduces to:
1
L = 𝑚𝑔𝑖𝑗 𝑞̇ 𝑖 𝑞̇ 𝑗
[5-11]
2
It can be shown that the above Lagrangian leads in
fact to the same Euler-Lagrange equations as the
Lagrangian relative to [5-9].
We can replace the minimization of J by that of the action:
𝐵
𝛿 ∫𝐴 𝑔𝑖𝑗 𝑞̇ 𝑖 𝑞̇ 𝑗 𝑑𝑢 = 0
[5-12]
And thereby simplifying the problem by eliminating the radical
(the square root).
The minimization in [5-12] is a relatively simple standard
problem in variational calculus.
Note that 𝑔𝑖𝑗 is a function of all the 𝑞 𝑘 but not on the
derivatives 𝑞̇ 𝑘 . There will be an Euler equation for each k:
𝜕𝑔𝑖𝑗 𝑞̇ 𝑖 𝑞̇ 𝑗
𝜕𝑞𝑘
−
𝑑
𝑑𝑢
𝜕𝑔𝑖𝑗 𝑞̇ 𝑖 𝑞̇ 𝑗
(
𝜕𝑞̇ 𝑘
Evaluate [5-13] to get:
)=0
[5-13]
23
𝜕𝑔𝑖𝑗
𝜕𝑞𝑘
𝑑
𝑖 𝑗
𝑞̇ 𝑞̇ −
𝑑𝑢
𝜕𝑔𝑖𝑗
𝑑
𝜕𝑞
𝑑𝑢
𝑖 𝑗
𝑘 𝑞̇ 𝑞̇ −
(𝑔𝑖𝑗
𝜕
𝑖 𝑗
(𝑞̇ 𝑞̇ )) = [
𝜕𝑞̇ 𝑘
𝜕𝑞̇ 𝑖
𝜕𝑞̇ 𝑗
= 𝛿 𝑖𝑗 ] =
(𝑔𝑘𝑗 𝑞̇ 𝑗 + 𝑔𝑖𝑘 𝑞̇ 𝑖 ) = 0
[5-14]
Simplify by using:
𝑑𝑞̇ 𝑗
𝑑𝑢
𝜕𝑔𝑘𝑗
= 𝑞̈ 𝑗 𝑎𝑛𝑑
𝑑𝑢
=
𝜕𝑔𝑘𝑗
𝑑𝑞 𝑖
𝑞̇ 𝑖 [𝑐ℎ𝑎𝑖𝑛 𝑟𝑢𝑙𝑒]
[5-15]
And [5-14] can be written as:
1
2
𝑞̇ 𝑖 𝑞̇ 𝑗 [
𝜕𝑔𝑖𝑗
𝜕𝑞𝑘
−
𝜕𝑔𝑘𝑗
𝜕𝑞𝑖
−
𝜕𝑔𝑖𝑘
𝜕𝑞
𝑖
]
−
𝑔
𝑞̈
=0
𝑖𝑘
𝑗
[5-16]
As a final simplification, multiply [5-16] by 𝑔𝑘𝑙 and use the
identity
𝜕2 𝑞𝑙
𝑑𝑢2
𝑔𝑘𝑙 𝑔𝑖𝑘 = 𝛿𝑖𝑙 to get the geodesic equation:
+
𝑑𝑞𝑖 𝑑𝑞𝑗 1
𝑑𝑢 𝑑𝑢 2
𝑔
𝑘𝑙
[
𝜕𝑔𝑘𝑗
𝜕𝑞𝑖
+
𝜕𝑔𝑖𝑘
𝜕𝑞𝑗
−
𝜕𝑔𝑖𝑗
𝜕𝑞𝑘
]=0
[5-17]
Or using the Christoffel symbols of the second kind written in
terms of the metric tensor:
𝜕2 𝑞𝑙
𝑑𝑢2
+
𝑑𝑞𝑖 𝑑𝑞𝑗
𝑑𝑢 𝑑𝑢
Γ𝑖𝑗𝑙 = 0
[5-18]
#3 Yet another way to derive the geodesic equation is to see
the geodesic as the curve with zero tangent acceleration. The
approach can be described as “take a parameterized curve and
let the space curvature, described by the metric tensor, move
all points along the curve to the correct positions”
Consider a curve
𝛼 = 𝛼(𝑡) = 𝑟(𝑥 (𝑡))
[5-19]
24
which is a sufficiently smooth function and
where 𝛼 ∶ 𝐼 → 𝑅 𝑁 , 𝐼 𝑖𝑠 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 [𝑡1 , 𝑡1 ]
Calling t the ‘time’, only a choice we make, we can call
𝛼̇ 𝑘 (𝑡) = 𝑟𝑘,𝑖 𝑥̇ 𝑖 (𝑡)
[5-20]
the velocity and
𝛼̈ 𝑘 (𝑡) = 𝑟𝑘,𝑖𝑗 𝑥̇ 𝑖 𝑥̇𝑗 + 𝑟𝑘,𝑖 𝑥̈ 𝑖
[5-21]
the acceleration.
The geodesic is the shape of the curve when the acceleration
has zero projection to the plane tangent to the given surface,
which gives the equations
𝛼̈ 𝑘 (𝑡) 𝑟𝑘,𝑛 = 0 𝑛 = 1 … 𝑁
[5-22]
and using 𝛼̈ 𝑘 (𝑡) = 𝑟𝑘,𝑖𝑗 𝑥̇ 𝑖 𝑥̇𝑗 + 𝑟𝑘,𝑖 𝑥̈ 𝑖
to get
(𝑟𝑘,𝑖𝑗 𝑥̇ 𝑖 𝑥̇𝑗 + 𝑟𝑘,𝑖 𝑥̈ 𝑖 )𝑟𝑘,𝑛 = 0
[5-21]
and just do the multiplication
𝑟𝑘,𝑖 𝑟𝑘,𝑛 𝑥̈ 𝑖 + 𝑟𝑘,𝑖𝑗 𝑟𝑘,𝑛 𝑥̇ 𝑖 𝑥̇𝑗 = 0
[5-23]
Before going on, some clarification of the meaning of the
condition stated in [5-22] :
It has the form 𝑎𝑘
𝑎⃗ ∙ 𝑑𝑟⃗ = 𝑎⃗ ∙
𝜕𝑟⃗
𝜕𝑥𝑛
𝜕𝑟𝑘
𝜕𝑥𝑛
, in vector form 𝑎⃗
∙
𝑑𝑥𝑛 which means that 𝑎𝑘
𝜕𝑟⃗
𝜕𝑥𝑛
𝜕𝑟𝑘
𝜕𝑥𝑛
,
= 0 gives
𝑎⃗ ⊥ 𝑑𝑟⃗ , the vector has no projection in the 𝑑𝑟⃗ direction.
Applied to [5-22] this means that 𝛼̈⃗ ⊥ 𝑑𝑟⃗ . Acceleration has
zero projection etc. as stated above. The acceleration, ‘force’, is
perpendicular to the curve and only changes the direction of the
curve in N-dim space. This condition is then expressed in terms
of the metric tensor for the space.
25
To get the geodesic expressed in terms of the metric tensor,
note the definition of the covariant metric tensor
𝑔𝑖𝑗 = 𝑟𝑚,𝑖 𝑟𝑚,𝑗 definition of the covariant metric tensor.
Derivation of the definition with respect to
𝑥𝑘
yields
𝑔𝑖𝑗,𝑘 = 𝑟𝑚,𝑖𝑘 𝑟𝑚,𝑗 + 𝑟𝑚,𝑖 𝑟𝑚,𝑗𝑘
[5-24]
and the two permutations of the indices
𝑔𝑗𝑘,𝑖 = 𝑟𝑚,𝑖𝑗 𝑟𝑚,𝑘 + 𝑟𝑚,𝑗 𝑟𝑚,𝑖𝑘
[5-25]
𝑔𝑘𝑖,𝑗 = 𝑟𝑚,𝑘𝑗 𝑟𝑚,𝑖 + 𝑟𝑚,𝑘 𝑟𝑚,𝑖𝑗
[5-26]
[5-24] + [5-25] - [5-26] gives :
𝑔𝑖𝑗,𝑘 + 𝑔𝑗𝑘,𝑖 − 𝑔𝑘𝑖,𝑗 = 2 𝑟𝑚,𝑖𝑘 𝑟𝑚,𝑗
[5-27]
Using the definition of the metric tensor + [5-27] into [5-23] :
1
𝑔𝑖𝑛 𝑥̈ 𝑖 + (𝑔𝑖𝑛,𝑗 + 𝑔𝑛𝑗,𝑖 − 𝑔𝑗𝑖,𝑛 )𝑥̇ 𝑖 𝑥̇𝑗 = 0
2
[5-28]
Multiplying with 𝑔𝑛𝑚 and noticing the identity 𝑔𝑖𝑛 𝑔𝑛𝑚 =
𝛿𝑖𝑚 , 𝑔𝑖𝑛 𝑥̈ 𝑖 + 12 (𝑔𝑖𝑛,𝑗 + 𝑔𝑛𝑗,𝑖 − 𝑔𝑗𝑖,𝑛 )𝑥̇ 𝑖 𝑥̇𝑗 = 0
[5-28] will
result in:
1
𝑥̈ 𝑚 + 𝑔𝑛𝑚 (𝑔𝑖𝑛,𝑗 + 𝑔𝑛𝑗,𝑖 − 𝑔𝑗𝑖,𝑛 )𝑥̇ 𝑖 𝑥̇𝑗 = 0
2
[5-29]
This is the geodesic equation and it can be written more
compact as before using the Christoffel symbol of the second
kind:
𝑥̈ 𝑚 + Γ𝑖𝑗𝑚 𝑥̇ 𝑖 𝑥̇𝑗 = 0
[5-30]
An Example “along the geodesic”:
Since the length along the geodesic is a scalar, the velocities
𝑑𝑞𝑖
𝑑𝑠
of a freely falling particle along the geodesic form a
26
contravariant vector. Hence 𝑉𝑘
𝜕𝑞 𝑘
𝑑𝑠
is a well-defined scalar on a
geodesic, which we can differentiate in order to define the
covariant derivative of any covariant vector 𝑉𝑘 .
𝑑
𝜕𝑞𝑘
𝑑𝑉𝑘 𝜕𝑞𝑘
𝑑2 𝑞𝑘
(𝑉
)=
+ 𝑉𝑘
𝑑𝑠 𝑘 𝑑𝑠
𝑑𝑠 𝑑𝑠
𝑑𝑠 2
= [𝑢𝑠𝑖𝑛𝑔 𝑔𝑒𝑜𝑑𝑒𝑠𝑖𝑐 𝑒𝑞. ]
𝑖
𝑗
𝜕𝑉𝑘 𝜕𝑞𝑖 𝜕𝑞𝑘
𝑘 𝜕𝑞 𝜕𝑞
=
− 𝑉𝑘 Γ𝑖𝑗
𝑖
𝜕𝑞 𝑑𝑠 𝑑𝑠
𝑑𝑠 𝑑𝑠
𝑖
𝑘
𝜕𝑞𝑖 𝜕𝑞𝑘 𝜕𝑉𝑘
𝜕𝑞
𝜕𝑞
𝑙
=
( 𝑖 − 𝑉𝑙 Γ𝑖𝑘
)=
𝑉
𝑑𝑠 𝑑𝑠 𝜕𝑞
𝑑𝑠 𝑑𝑠 𝑘;𝑖
The quotient theorem tells us that 𝑉𝑘;𝑖 is a covariant tensor
that defines the covariant derivative of 𝑉𝑘 Similarly, higherorder tensors may derived.
Some concluding remarks:
The Mass and Space ‘marriage’, stolen from somewhere,
“Mass tells space how to curve and curved space tells
mass how to move”.
And as a reminder that there is always more to learn,
Einstein’s field equations that are still researched.
This article hopefully gives a hint to understand some of the
terms :
1
8πG
𝑅𝜇𝜈 − 𝑔𝜇𝜈 𝑅 + 𝑔𝜇𝜈 Λ = 4 𝑇𝜇𝜈
2
𝑐
where 𝑅𝜇𝜈 is the Ricci curvature tensor,
the scalar
𝑔𝜇𝜈 the metric tensor, is the cosmological
constant, G is Newton's gravitational constant, c the speed of
light in vacuum, and 𝑇𝜇𝜈 the stress–energy tensor.
curvature,
27
6.
Appendix A, a wakeup problem
Living in flat Eucledian space ?
The following example serves as an illustration of the
importance of the geometry of space itself and the importance
of choosing a proper coordinate system.
Here is the problem:
We have a box whose short sides are squares,
1.2 meters x 1.2 meters.
The length of the box is 3 meters.
In the middle of one of the short sides, 0.1 meter from the top
side of the box is a spider.
In the middle of the other short side, 0.1 meter from the
bottom side of the box, is a fly, caught in the spider’s web.
The spider wants to catch the fly as fast as possible.
The spider can only move on the surface of the box.
The geometry is strictly Euclidean on the surfaces of the box,
but with ‘singularities’ at the edges.
How long is the shortest path from the spider to the fly ?
0.1 m
1.2 m
0.1 m
3.0 m
1.2 m
28
Yes, with the proper “coordinate
transformation”
By unfolding the box in different ways, the problem is solved
using a straight ruler, the Pythagorean theorem and some
thinking.
The easy answer is 4.2 meters, but there are more straight
lines to the prey:
Four different unfolding are shown, three giving values # 4.2
meters, two of them shorter than 4.2 meters:
#1 the easy one
#2 the no-good one
29
#3 shorter than the easy one
#4 the shortest path, chosen by the spider that has mirrors
helping him to see round all the edges
This example shows the intrinsic nature of curved space,
where care must be taken in defining the notion of “shortest
path”. In general curved space the curvature is different in each
point in the space, but the space is normally smooth, no ‘edges’
like in this example.
Some mapping notes
Different mapping methods can simplify the mathematics of a
problem considerably. In high power microwave transmission it
is common to use pipes with rectangular cross section and
30
conformal mapping can be used to get a circular cross
section, solve the problem in that geometry and then use the
inverse map to get the result for the rectangular cross section.
Riemann’s mapping theorem shows that almost any area in
the complex plane can be mapped to the interior of the unit
circle. Riemann’s inverse theorem is about mapping to the area
outside the unit circle. With some “minimal” conditions the map
can be shown to be unique. The conditions for the mapping is
“a non-empty simply connected open subset of the complex
number plane which is not all of , then there exists a
biholomorphic (bijective and holomorphic) mapping from
onto the open unit disk”.
Another example of using mapping techniques is the laminar
flow around the wing of an airplane. The wing’s cross-section
is mapped to a circle. This is probably not used anymore with
access to today’s computer power.
31
7.
Appendix B, index notation
Information about index notation
Index notation is used extensively in tensor algebra (and
thereby in general relativity theory), differential geometry,
matrices, determinants and more. A basic understanding is
needed to read "more advanced" physics textbooks and
understanding some basics about index notation is a good and
simple-to-learn tool.
𝑎⃗ has components 𝑎𝑖 , and a vector 𝑏⃗⃗ has
⃗⃗
components 𝑏𝑖 , the scalar product of the two vectors is 𝑎⃗∙ 𝑏
= ∑𝑖 𝑎𝑖 𝑏𝑖 = 𝑎𝑖 𝑏𝑖 if Einstein's summation convention is used,
A vector
meaning that anytime two same indices appear it is an implicit
summation over that index.
NB! writing 𝑎⃗ = 𝑎𝑖 is a misuse of the = sign, since right hand
side is a scalar and left hand side is a vector.
NB! Einstein’s convention cannot always be used, e.g.
𝜀𝑖𝑗𝑘 𝐵𝑖𝑗 𝐶𝑖 is meaningless, Σ has to be used. Also e.g. if
readability is compromised. Use with common sense.
𝑎⃗ and 𝑏⃗⃗ can be written as
⃗⃗ = 𝑎𝑖 𝒆
⃗⃗𝒊 and ⃗𝒃⃗ = 𝑏𝑖 𝒆
⃗⃗𝒊 , which gives the scalar product
𝒂
⃗⃗𝒊 𝑏𝑘 𝒆
⃗⃗𝒌 = 𝑎𝑖 𝑏𝑘 𝒆
⃗⃗𝒊 𝒆
⃗⃗𝒌 . This is a large number of terms,
𝑎𝑖 𝒆
The two vectors
summing over i=1…n and k=1…n, quite far from the nice
formula in the preceding paragraph !
The Kronecker delta is defined as 𝛿𝑖𝑗
𝑗
= 𝛿𝑗𝑖 = 𝛿𝑖 = 0 if i # j
and = 0 if i = j
The condition for orthonormal basis vectors can be written as
⃗⃗𝒊 ∙ 𝒆
⃗⃗𝒋 = 𝛿𝑖𝑗 ,
𝒆
which gives
⃗⃗𝒊 𝑏𝑘 𝒆
⃗⃗𝒌 = 𝑎𝑖 𝑏𝑘 𝒆
⃗⃗𝒊 ∙ 𝒆
⃗⃗𝒌 = 𝑎𝑖 𝑏𝑘 𝛿𝑖𝑘 =
𝑎⃗∙𝑏⃗⃗ = 𝑎𝑖 𝒆
32
{ k can be set to i, all other terms are multiplied with 0, notice
that when doing this contraction care must be taken which
indices are just dummies for e.g. a summation or which one has
other semantic content }
= 𝑎𝑖 𝑏𝑖 , the nice expression you usually see.
NB! 𝛿𝑖𝑖 = 𝛿11 + 𝛿22 + 𝛿33 = 3 (= N if N dimensions)
Can one avoid the mess caused by general coordinate
systems where the basis vectors are neither orthogonal nor
normalized ?
Yes, with the basis vectors chosen in a smart way. Then the
⃗⃗ will look like 𝑎⃗∙𝑏⃗⃗ = 𝑎𝑖 𝑏 𝑖 where the
scalar product of 𝑎⃗ and 𝑏
sub/lower index denotes covariant vector components and the
super/upper index contra variant vector components.
𝑎⃗∙𝑏⃗⃗ = 𝑎𝑖 𝑏 𝑖 = 𝑎𝑖 𝑏𝑖 for a general curvilinear coordinate
system.
Another important symbol used is the Levi-Cevita symbol,
sometimes also called the permutation symbol. It is denoted e
or 𝜀 in the case that be misunderstood as Euler’s number e.
The definition in 3 dimensions, i,j,k=1..3, is 𝜀𝑖𝑗𝑘 { = 0 if two of
i, j, k are equal, = 1 if i , j, k is an even permutation, = -1 if i, j, k
is an odd permutation }
The definition of 𝜀𝑖𝑗𝑘 is closely coupled to the definition of the
determinant, which is 0 if two rows or columns are equal and
which changes sign if two rows or columns changes sign.
Consequently, with a matrix A with elements 𝑎𝑖𝑗 , the
determinant |𝑎𝑖𝑗 | = 𝜀𝑖𝑗𝑘 𝑎1𝑖 𝑎2𝑗 𝑎3𝑘 = 𝜀𝑖𝑗𝑘 𝑎𝑖1 𝑎𝑗2 𝑎𝑘3 ,
where on the right hand side the i, j, k are just dummy
summation indices 1..3, on the left hand side the i, j picks the
element in the matrix.
NB! that 𝜀𝑖𝑗𝑘 𝜀𝑙𝑚𝑛
= 3! (= N! if N dimensions)
Using this with matrix A with elements 𝑎𝑖𝑗 , the determinant
33
|𝑎𝑖𝑗 | = 𝜀𝑖𝑗𝑘 𝑎1𝑖 𝑎2𝑗 𝑎3𝑘 =
1
1
𝜀 𝜀 𝜀 𝑎 𝑎 𝑎
3! 𝑖𝑗𝑘 𝑙𝑚𝑛 𝑙𝑚𝑛 1𝑖 2𝑗 3𝑘
=
𝜀 𝜀 𝑎 𝑎 𝑎
3! 𝑖𝑗𝑘 𝑙𝑚𝑛 𝑖𝑙 𝑗𝑚 𝑘𝑛
The generalized Kronecker delta is defined as
“generalized delta”
𝑖𝑗𝑘
𝛿𝑙𝑚𝑛
𝛿𝑙𝑖
𝑖
𝛿𝑚
= | 𝛿𝑙𝑗
𝛿𝑚
𝛿𝑛 |
𝛿𝑙𝑘
𝑘
𝛿𝑚
𝛿𝑛𝑘
𝑗
𝛿𝑛𝑖
𝑗
Using the generalized Kronecker delta, we can get the very
useful epsilon-delta identity (“𝜀 – 𝛿 identity”), here is how:
𝛿11 𝛿21 𝛿31
1 0 0
1 = |0 1 0| = |𝛿12 𝛿22 𝛿32 | , 𝜀 𝑖𝑗𝑘 =
0 0 1
𝛿13 𝛿23 𝛿33
𝛿1𝑖 𝛿2𝑖 𝛿3𝑖
| 𝛿1𝑗 𝛿2𝑗 𝛿3𝑗 | “row shift taken care of by 𝜀 𝑖𝑗𝑘 ” =
𝛿1𝑘
𝛿2𝑘
𝜀 𝑖𝑗𝑘 𝜀𝑙𝑚𝑛
𝛿3𝑘
𝛿𝑙𝑖
= | 𝛿𝑙𝑗
𝑖
𝛿𝑚
𝛿𝑚
𝛿𝑛 |
𝛿𝑙𝑘
𝑘
𝛿𝑚
𝛿𝑛𝑘
𝑗
𝛿𝑛𝑖
𝑗
“ column shift taken care of by
𝜀𝑙𝑚𝑛 ”
Now, do a contraction by setting i = l and evaluate the
determinant on the right hand side using well known algorithms
to get:
𝑗
𝑗
𝑘
“𝜀 – 𝛿 identity” : 𝜀 𝑖𝑗𝑘 𝜀𝑖𝑚𝑛 = 𝛿𝑚 𝛿𝑛𝑘 - 𝛿𝑛 𝛿𝑚
There is a special notation for writing derivatives :
Example 1, j-component of ∇𝜙
: 𝑒⃗𝐣 ∙ ∇𝜙 = 𝜙,𝑗 =
𝜕Φ
𝜕𝑥 𝑗
34
Example 2, second partial derivative :
𝜙,𝑗𝑘 =
𝜕2 Φ
𝜕𝑥 𝑗 ð𝑥 𝑘
NB! A shorthand for partial derivative can also be 𝜕𝑘
meaning “partial derivative in the k direction”.
=
𝜕
𝜕𝑥𝑘
,
Examples index notation, Cartesian coordinates
Example 3:
𝑎⃗ × 𝑏⃗⃗ = 𝜀𝑖𝑗𝑘 𝑎𝑗 𝑏𝑘 𝑒⃗𝑖
Example 4:
𝑎⃗ ∙ (𝑏⃗⃗ × 𝑐⃗) = 𝜀𝑖𝑗𝑘 𝑎𝑖 𝑏𝑗 𝑐𝑘
∇ × (𝑎⃗ × 𝑏⃗⃗) = 𝑎⃗ (∇ ∙ 𝑏⃗⃗) − ⃗⃗⃗⃗
𝑏 (∇ ∙ 𝑎⃗) +
(𝑏⃗⃗ ∙ ∇)𝑎⃗ − (𝑎⃗ ∙ ∇)𝑏⃗⃗ “not too easy ?”
Example 5:
Using index notation: take the 𝑒⃗𝑝 component of the vector
[often used this way] =>
𝑒⃗𝑝 ∙ [∇ × (𝑎⃗ × 𝑏⃗⃗)] = 𝜀𝑝𝑞𝑘 [𝜀𝑘𝑗𝑖 𝑎𝑗 𝑏𝑖 ],𝑞 = [derivative of a
product] = 𝜀𝑝𝑞𝑘 𝜀𝑘𝑗𝑖 [𝑎𝑗 𝑏𝑖,𝑞 + 𝑎𝑗,𝑞 𝑏𝑖 ] = 𝜀𝑝𝑞𝑘 𝜀𝑘𝑗𝑖 𝑎𝑗 𝑏𝑖,𝑞 +
𝜀𝑝𝑞𝑘 𝜀𝑘𝑗𝑖 𝑎𝑗,𝑞 𝑏𝑖 = [ “epsilon-delta identity” ] =
[𝛿𝑝𝑗 𝛿𝑞𝑖 - 𝛿𝑝𝑖 𝛿𝑞𝑗 ] 𝑎𝑗 𝑏𝑖,𝑞 + [𝛿𝑝𝑗 𝛿𝑞𝑖 - 𝛿𝑝𝑖 𝛿𝑞𝑗 ] 𝑎𝑗,𝑞 𝑏𝑖 =
[use 𝛿 properties] =
𝑎𝑝 𝑏𝑘,𝑘 - 𝑎𝑘 𝑏𝑝,𝑘 + 𝑎𝑝,𝑘 𝑏𝑘 - 𝑎𝑘,𝑘 𝑏𝑝 And the vector identity
⃗⃗ 𝑎𝑘,𝑘 =∇ ∙ 𝑎⃗
can be easily recognized e.g. 𝑏𝑘,𝑘 =∇ ∙ 𝑏
𝑎𝑘 𝑏𝑝,𝑘 = 𝑒⃗𝑝 ∙ (𝑎⃗ ∙ ∇) 𝑏⃗⃗))
Example 6: ∇ × (∇∅) = ⃗0⃗ “Curl of a gradient field is the
zero vector”. Taking the i component of the vector using index
notation
𝜀𝑖𝑗𝑘
𝜕 𝜕Φ
𝜕𝑥𝑗 𝜕𝑥𝑘
= 𝜀𝑖𝑗𝑘
𝜕2 Φ
𝜕𝑥𝑗 𝜕𝑥𝑘
, with i fixed this results in two
35
terms 𝜀𝑖𝑗𝑘
𝜕2 Φ
𝜕2 Φ
𝜕𝑥𝑗 𝜕𝑥𝑘
+ 𝜀𝑖𝑘𝑗
𝜕2 Φ
𝜕𝑥𝑘 𝜕𝑥𝑗
j,k # i = 𝜀𝑖𝑗𝑘
𝜕2 Φ
𝜕𝑥𝑗 𝜕𝑥𝑘
- 𝜀𝑖𝑗𝑘
j,k # i = 0 (if it is OK to change order of derivatives)
𝜕𝑥𝑘 𝜕𝑥𝑗
∇ ∙ (∇ × 𝐹⃗ ) = 0 “Divergence of a curl is zero”
𝜕
𝜕𝐹
Using index notation ∇ ∙ (∇ × 𝐹⃗ ) =
𝑒⃗𝑚 ∙ 𝜀𝑖𝑗𝑘 𝑘 𝑒⃗𝑖 =
Example 7:
𝜕𝑥𝑚
𝜕𝑥𝑗
𝜕 𝜕𝐹𝑘
𝜕 𝜕𝐹𝑘
𝜀𝑖𝑗𝑘
𝑒⃗𝑚 ∙ 𝑒⃗𝑖 = 𝜀𝑖𝑗𝑘
𝛿 =
𝜕𝑥𝑚 𝜕𝑥𝑗
𝜕𝑥𝑚 𝜕𝑥𝑗 𝑖𝑚
𝜕 2 𝐹𝑘
𝜕 2 𝐹𝑘
𝜀𝑖𝑗𝑘
𝛿𝑖𝑚 = 𝜀𝑖𝑗𝑘
= [𝜀𝑖𝑗𝑘 def. + changing order
𝜕𝑥 𝜕𝑥
𝜕𝑥 𝜕𝑥
𝑚
𝑗
𝑖
𝑗
of derivation=no change] = 0
Example 8: Matrix multiplication
A matrix A with elements 𝑎𝑖𝑗
elements 𝑥𝑖 i = 1..3
⃗ with
i,j = 1..3 and a vector 𝑋
A∙ 𝑋⃗ = B will be a 3x1 matrix with elements 𝑏𝑖 = 𝑎𝑖𝑗 𝑥𝑗
Example 9: Let A be an mxn matrix with elements
𝑎𝑖𝑗 i = 1..m, j=1..n
B an nxp matrix with elements
𝑏𝑘𝑙 k = 1..n, k=1..p
A∙B = C will be an mxp matrix with elements
𝑐𝑖𝑗 = 𝑎𝑖𝑘 𝑏𝑘𝑗 , k=1..n , i=1..m, j=1..p
36
8.
Appendix C, remarks of interest
No coordinate system ?
Differential forms are an approach to multivariable calculus
that is independent of coordinates
(http://en.wikipedia.org/wiki/Differential_form).
The differential-form framework has brought considerable
unification to vector algebra and to tensor analysis on manifolds
more generally…[Arfken].
Maxwell’s homogenous equations can be formulated as simply
dF = 0 and his inhomogeneous equations take the elegant form
d(*F) = *J
Adding dimensions, topology
In April 1919 Theodor Kaluza noticed that when he solved
Albert Einstein's equations for general relativity using five
dimensions, then James Clark Maxwell's equations for
electromagnetism emerged spontaneously. Kaluza wrote to
Einstein who, in turn, encouraged him to publish. Kaluza's
theory was published in 1921 in a paper, "Zum Unitätsproblem
der Physik" with Einstein's support. It is now called Kaluza–
Klein theory (KK theory) and is a model that seeks to unify the
two fundamental forces of gravitation and electromagnetism.
Klein is known among other things for his ‘bottle’.
Klein bottle is a non-orientable surface; informally, it can be
defined as a surface (a two-dimensional manifold) in which
notions of left and right cannot be consistently defined. Other
related non-orientable objects include the Möbius strip and the
real projective plane. Whereas a Möbius strip is a surface with
boundary, a Klein bottle has no boundary (for comparison, a
sphere is an orientable surface with no boundary).
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The Klein bottle.
Beauty ?
Illustrations of hyperbolic geometry by M.C. Escher.
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9.
Appendix D, T-shirt from CERN
This confusing T-shirt from CERN demonstrates how complex
physical laws can be written in a simple way and make it
possible to write equations from different fields of physics in a
similar form.
The leaflet that came with the shirt says :
This equation neatly sums up our current understanding of
fundamental particles and forces.
It represents mathematically what we call the standard model
of particle physics.
The top line describes the forces: electricity, magnetism and
the strong and weak nuclear forces.
The second line describes how these forces act on the
fundamental particles of matter, namely the quarks and leptons.
The third line describes how these particles obtain their
masses from the Higgs boson.
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The fourth line enables the Higgs boson to do the job.
Many experiments at CERN and other laboratories have
verified the top two lines in detail.
One of the primary objectives of the LHC was to see whether
the Higgs boson exists, now confirmed, and behaves as
predicted by the last two lines.
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10. References
[Arfken] Mathematical Methods for Physicists, Georg B.
Arfken , Hans J. Weber and Frank E. Harris , 6th and 7th edition
[Heinbockel] Professor J.H. Heinbockel Introduction to tensor
calculus and Continuum Mechanics
[Waleffe] Article available on internet about curvilinear
coordinates by Professor Fabian Waleffe, University of
Wisconsin-Madison
[Penrose] The Road to Reality: A Complete Guide to the
Laws of the Universe, Sir Roger Penrose
[Wikipedia] http://en.wikipedia.org
[Youtube] http://youtube.com/mathview , differential geometry
and more