44
4 Riemann Spaces and Tensors
To obtain the basis 1-forms we may use the transformation rule (4.27) since the
upper index position determines the transformation; alternatively we may use
the demand that the forms obey the orthogonality relation (4.30). The result is
dr = cos ϕ dx + sin ϕ dy, dϕ = −
1
ρ
(sin ϕ dx + cos ϕ dy).
(4.37)
We will later further discuss the normalization of the 1-forms.
4.4 Tensors, Component View
We continue in this section with the classic component view of vectors and tensors
as indexed arrays. This section consists largely of a set of theorems which are proved
by a relatively simple algebraic process often called index juggling. It should become
clear that after some practice the balancing of the indices does much of the work for
us.
To define a tensor we generalize the idea of a vector as defined as an n-tuple with
a well-defined transformation between coordinate systems: a tensor is defined as a
set of quantities with any number of indices, which transforms according to
T
l...
m... =
∂ x
l
∂ x q . . .
∂ x
n
∂ x
m . . . T
q...
n... , tensor components.
(4.38)
The total number of indices is referred to as the rank; some of the indices may be
upper, or contravariant, and others may be lower, or covariant. The number of such
indices is written as (M, N). Thus for example a vector is a first rank tensor and (1,
0). Another example is V
j W q , which is a second rank tensor and (1, 1).
From this tensor definition many simple but powerful theorems follow. We have
already introduced and proved two of them in Sect. 4.2: Theorem 1 concerned the
Jacobian matrices and Theorem 2 the invariance of the inner product of vectors. Let
us continue to more such theorems.
Theorem 3 To contract a tensor we set an upper index equal to a lower index
and sum, which gives another tensor; for example one contraction of T
αβ λγ is
T
αβ βγ = S
α γ . Contraction of a rank r tensor produces a rank r − 2 tensor. Consider
the above 4th rank tensor as an example. Then the contracted object transforms as
T
αβ
βγ =
∂ ¯
x
α
∂ x ω
∂ ¯
x
β
∂ x σ
∂ x
λ
∂ ¯
x β
∂ x
η
∂ ¯
x γ T
ωσ λη =
∂ ¯
x
α
∂ x ω δ
λ
σ
∂ x
η
∂ ¯
x γ T
ωσ λη
=
∂ ¯
x
α
∂ x ω
∂ x
η
∂ ¯
x γ T
ωσ σ η .
(4.39)
4 Riemann Spaces and Tensors
To obtain the basis 1-forms we may use the transformation rule (4.27) since the
upper index position determines the transformation; alternatively we may use
the demand that the forms obey the orthogonality relation (4.30). The result is
dr = cos ϕ dx + sin ϕ dy, dϕ = −
1
ρ
(sin ϕ dx + cos ϕ dy).
(4.37)
We will later further discuss the normalization of the 1-forms.
4.4 Tensors, Component View
We continue in this section with the classic component view of vectors and tensors
as indexed arrays. This section consists largely of a set of theorems which are proved
by a relatively simple algebraic process often called index juggling. It should become
clear that after some practice the balancing of the indices does much of the work for
us.
To define a tensor we generalize the idea of a vector as defined as an n-tuple with
a well-defined transformation between coordinate systems: a tensor is defined as a
set of quantities with any number of indices, which transforms according to
T
l...
m... =
∂ x
l
∂ x q . . .
∂ x
n
∂ x
m . . . T
q...
n... , tensor components.
(4.38)
The total number of indices is referred to as the rank; some of the indices may be
upper, or contravariant, and others may be lower, or covariant. The number of such
indices is written as (M, N). Thus for example a vector is a first rank tensor and (1,
0). Another example is V
j W q , which is a second rank tensor and (1, 1).
From this tensor definition many simple but powerful theorems follow. We have
already introduced and proved two of them in Sect. 4.2: Theorem 1 concerned the
Jacobian matrices and Theorem 2 the invariance of the inner product of vectors. Let
us continue to more such theorems.
Theorem 3 To contract a tensor we set an upper index equal to a lower index
and sum, which gives another tensor; for example one contraction of T
αβ λγ is
T
αβ βγ = S
α γ . Contraction of a rank r tensor produces a rank r − 2 tensor. Consider
the above 4th rank tensor as an example. Then the contracted object transforms as
T
αβ
βγ =
∂ ¯
x
α
∂ x ω
∂ ¯
x
β
∂ x σ
∂ x
λ
∂ ¯
x β
∂ x
η
∂ ¯
x γ T
ωσ λη =
∂ ¯
x
α
∂ x ω δ
λ
σ
∂ x
η
∂ ¯
x γ T
ωσ λη
=
∂ ¯
x
α
∂ x ω
∂ x
η
∂ ¯
x γ T
ωσ σ η .
(4.39)