46
4 Riemann Spaces and Tensors
Because of (4.42) these are equal. The general case of any tensor equation is clear
from this example. This theorem is the basis of a very powerful method of proof for
tensor equations: they need only be proved in one convenient coordinate system, and
are then automatically true in any coordinate system. An equation of this type between
tensors is called form invariant, or covariant since the two sides vary together under
the transformation.
Theorem 9 Any tensor can be expanded as the sum of outer products of vectors.
As usual we illustrate this with a special case, a second rank (2, 0) tensor T
μν . To
construct the expansion in an n-dimensional space choose one coordinate system,
and in that system set up n contravariant vectors defined by U
α
j = δ
α
j , where j labels
which vector and α labels the components. Next, in that coordinate system define a
set of scalars equal to the components of the tensor, that is a i j = T
i j . Then obviously,
by construction
T
αβ
=
n
i, j
a i j δ
α
i δ
β
j =
n
i, j
a i j U
α
i U
β
j .
(4.44)
We have thereby constructed the desired expansion in the chosen coordinate system.
Most important, we have defined the U
α
J to be vectors and the a i j to be scalars, with
values given in the chosen system and defined in another system by the transformation
laws. Thus (4.42) is a tensor equation and true in any coordinate system, so the
expansion is generally valid. As might be expected this theorem is often useful in
proving other theorems. We have proven it using n vectors in n-dimensional space;
this is not the minimum number of vectors in the tensor expansion, but suffices to
prove most theorems of interest.
Theorem 10 (The Quotient Theorem) Suppose we have an array T
αβ and we are
given that for any vector V β the array
S
α
= T
αβ V β ,
(4.45)
is a vector; then the array T
αβ must be a tensor.
To prove this theorem express S
α and V β in terms of vectors in the barred system,
and write the above as
∂ x
α
∂ x
λ
S
λ = T
αβ
∂ x
τ
∂ x β V τ
.
(4.46)
Now multiply and contract both sides of this with ∂ x
ω
/∂ x
α and use Theorem 1 to
obtain
∂ x
ω
∂ x α
∂ x
α
∂ x
λ
S
λ = S
ω =
T
αβ ∂ x
ω
∂ x α
∂ x
τ
∂ x β V τ
= T
ωτ V τ ,
(4.47)
4 Riemann Spaces and Tensors
Because of (4.42) these are equal. The general case of any tensor equation is clear
from this example. This theorem is the basis of a very powerful method of proof for
tensor equations: they need only be proved in one convenient coordinate system, and
are then automatically true in any coordinate system. An equation of this type between
tensors is called form invariant, or covariant since the two sides vary together under
the transformation.
Theorem 9 Any tensor can be expanded as the sum of outer products of vectors.
As usual we illustrate this with a special case, a second rank (2, 0) tensor T
μν . To
construct the expansion in an n-dimensional space choose one coordinate system,
and in that system set up n contravariant vectors defined by U
α
j = δ
α
j , where j labels
which vector and α labels the components. Next, in that coordinate system define a
set of scalars equal to the components of the tensor, that is a i j = T
i j . Then obviously,
by construction
T
αβ
=
n
i, j
a i j δ
α
i δ
β
j =
n
i, j
a i j U
α
i U
β
j .
(4.44)
We have thereby constructed the desired expansion in the chosen coordinate system.
Most important, we have defined the U
α
J to be vectors and the a i j to be scalars, with
values given in the chosen system and defined in another system by the transformation
laws. Thus (4.42) is a tensor equation and true in any coordinate system, so the
expansion is generally valid. As might be expected this theorem is often useful in
proving other theorems. We have proven it using n vectors in n-dimensional space;
this is not the minimum number of vectors in the tensor expansion, but suffices to
prove most theorems of interest.
Theorem 10 (The Quotient Theorem) Suppose we have an array T
αβ and we are
given that for any vector V β the array
S
α
= T
αβ V β ,
(4.45)
is a vector; then the array T
αβ must be a tensor.
To prove this theorem express S
α and V β in terms of vectors in the barred system,
and write the above as
∂ x
α
∂ x
λ
S
λ = T
αβ
∂ x
τ
∂ x β V τ
.
(4.46)
Now multiply and contract both sides of this with ∂ x
ω
/∂ x
α and use Theorem 1 to
obtain
∂ x
ω
∂ x α
∂ x
α
∂ x
λ
S
λ = S
ω =
T
αβ ∂ x
ω
∂ x α
∂ x
τ
∂ x β V τ
= T
ωτ V τ ,
(4.47)