Master Thesis

One can see, that the differentials build a vector space too, which is named cotangent space or dual space. In
particular, the component functions x i : M 4 →�, x i(x 1, x 2, x 3, x 4)= x i satisfy
(d xi)(∂ j)=∂ j x i =〈d x i ,∂ j〉=δ i
j , (3.7)
which is the reason, why the d x i are said to be the basis of the dual base T∗ p (M). The corresponding ‘vector’ in the dual
spaceω=ω id x i∈ T∗ p (M) is called a one-form.
Definition 3.2.3 (Vector Field). Let p∈ M be a point of a manifold M. The mapping X : p→X p is called a vector field
on M. The set of all smooth (C ∞ ) vector fields on M is denoted by�(M). The set of all smooth mappingsω : p→ω p of
one-forms on M is denoted by� ∗ (M)
In local coordinates(x 1, x 2, x 3, x 4) of an open subset U∈M,�(M) and� ∗ (M) have the representations:
3.3 Tensors andtheir transformationbehavior
X p=ξ i �
∂
(p)
∂ x i
�
∈�(M) ωp=ζ i(p)
p
� d x i�
p ∈�∗ (M) (3.8)
i ∂
Let X= X
∂ x i∈�(M) be a vector andω=ω id x i∈� ∗ (M) a one-form. Under the change of coordinates given by the
mapping x ′ : M 4→ N 4 , x ′i (x 1 , x2 , x3 , x4 ), x ′ (p)= q the vectors transform in the following way:
X .x i′
k′ ∂ j i′
= X x = X
j∂ x
j ∂ x ∂ x jδi′
ω( ∂ ∂
∂ x i′)= ∂ x i′ω jd x j =ω j
∂ x j
∂ x
k′δ k′
i
k′ i′ x
k ′=∂ X
j∂x
j ∂ x ∂ x k′ = X
j
∂ x
′=
∂ x k′ω ∂ x
j
k′
∂ x i′ = ∂
∂ x i′ω k ′d x k′
k′ ∂
∂ x k′ x i′
(3.9)
(3.10)
By comparing the coefficients one comes to the conclusion, that the components of the vectors and one-forms transform
with the Jacobian of the coordinate change, also named tangent map
X i′
= ∂ x i′
j
X
∂ x j
,
∂ i′
X= X
∂ x i′∈�(N)
(3.11)
j ∂ x
ωi ′=
∂ x i′ω j , ω=ω i ′d x i′
∈� ∗ (N) . (3.12)
The tensorial product of r vectors Y a= Y i a ∂ia ∈�(M) and s one-formsΩ b =Ω jb d x j b ∈� ∗ (M) leads to an object:
T = Y 1⊗ . . .⊗ Y r⊗Ω 1 ⊗ . . .⊗Ω s = (3.13)
= T i1...ir j ∂i1⊗ . . .⊗∂ ir⊗ d x
1...j s j1 js ⊗ . . .⊗ d x (3.14)
T i1...ir j1...j s
= Y i 1 · . . .· Y i r·Ωj1 · . . .·Ω js , (3.15)
which is transformed component-wise by the tangential mapping of the coordinate change:
T i′
1 ...i′ r
j ′
1 ...j′ =
s
∂ x i′
1
∂ x k · . . .·
1
∂ x i′
r
∂ x k r
· ∂ x l1 ∂ x j′
· . . .·
1
∂ x ls ∂ x j′ · T
s
k1...kr l1...ls (3.16)
As a result of this transformation properties, T∈� r
s (M) is called a tensor field of type (r,s) (contra-variant of rank r,
covariant of rank s).
The numbers T i1...ir j need not be a product of components of vectors, as it is done in equation (3.15). The∂ i1⊗ . . .⊗
1...j s
∂ ir ⊗ d x j 1⊗ . . .⊗ d x j s are then the dim(� r
s (M))=nr+s basis elements of the cartesian product� r
s (M)= T pM× . . .× T pM×
T ∗
p
M× . . .× T∗
p
M of the tensor space.
Given a linear transformation Ai′ r
j , tensors T∈� s (M) transform in general by:
A(T) : T i′
1 ...i′
r
j ′
1 ...j′ s
= A i′
1
k 1
· . . .·A i′
r
k r
· A l1 j ′ · . . .·A
1
ls j ′ · T
s
k1...kr l1...ls (3.17)
10 3.3 Tensors andtheirtransformation behavior