48
4 Riemann Spaces and Tensors
When we lower or raise an index we usually retain the same name for the tensor;
hence we may think of a tensor as being expressible in terms of its contravariant
components or covariant components or any combination of the 2. This is in accord
with the abstract view we will address in the next section.
From the defining relation of the inverse metric in (4.50) the Kronecker delta is
the mixed metric tensor so we may express it as g
α β = δ
α
β ; it is an unusual tensor in
that it is the same in all coordinate systems as we noted above.
4.5 Tensors, Abstract View
As with vectors we may view tensors as abstract objects instead of from the classic
component point of view discussed in the previous section. In this abstract approach
an (M, N) tensor is defined to linearly map M vectors and N 1-forms to the reals. For
example a (0, 2) tensor T operates as a linear map on vectors
V ,
W as follows
T
V ,
W
= T
V
β
e β , W
μ
e μ
= V
β W
μ T
e β ,
e μ
≡ V
β W
μ T βμ
(4.54)
The components T βμ defined here are the same as we discussed in the previous
section. Thus a vector is also a (1, 0) tensor and a 1-form is also a (0, 1) tensor. The
metric is the most important special case of a (0, 2) tensor, so we explicitly note its
operation in terms of components
g
V ,
W
= V
β W
μ g βμ
(4.55)
Let’s look at another example of a (0, 2) tensor. Define the direct product of
two 1-forms as something that operates linearly on two vectors to give a real in the
following natural way,
p ⊗ q = direct product of 1-forms, p ⊗ q
V ,
W
= p
V
q
W
(4.56)
That is, the first factor in the direct product operates on the first vector and the second
factor in the direct product operates on the second vector. The direct product in (4.56)
is thus a (0, 2) tensor. It should be clear that we can extend the definition to the direct
product of any number M of 1-form factors to produce a (0, M) tensor and so forth.
Recall that we discussed in Sect. 4.3 a basis for 1-forms which we denoted as ω
α .
We can similarly show that there exists a basis for the product of two 1-forms or (0,
2) tensors. Indeed the basis is a linear combination of the direct product of the ω
α .
We write that linear combination as
f = f αβ ω
α
⊗ ω
β
.
(4.57)
4 Riemann Spaces and Tensors
When we lower or raise an index we usually retain the same name for the tensor;
hence we may think of a tensor as being expressible in terms of its contravariant
components or covariant components or any combination of the 2. This is in accord
with the abstract view we will address in the next section.
From the defining relation of the inverse metric in (4.50) the Kronecker delta is
the mixed metric tensor so we may express it as g
α β = δ
α
β ; it is an unusual tensor in
that it is the same in all coordinate systems as we noted above.
4.5 Tensors, Abstract View
As with vectors we may view tensors as abstract objects instead of from the classic
component point of view discussed in the previous section. In this abstract approach
an (M, N) tensor is defined to linearly map M vectors and N 1-forms to the reals. For
example a (0, 2) tensor T operates as a linear map on vectors
V ,
W as follows
T
V ,
W
= T
V
β
e β , W
μ
e μ
= V
β W
μ T
e β ,
e μ
≡ V
β W
μ T βμ
(4.54)
The components T βμ defined here are the same as we discussed in the previous
section. Thus a vector is also a (1, 0) tensor and a 1-form is also a (0, 1) tensor. The
metric is the most important special case of a (0, 2) tensor, so we explicitly note its
operation in terms of components
g
V ,
W
= V
β W
μ g βμ
(4.55)
Let’s look at another example of a (0, 2) tensor. Define the direct product of
two 1-forms as something that operates linearly on two vectors to give a real in the
following natural way,
p ⊗ q = direct product of 1-forms, p ⊗ q
V ,
W
= p
V
q
W
(4.56)
That is, the first factor in the direct product operates on the first vector and the second
factor in the direct product operates on the second vector. The direct product in (4.56)
is thus a (0, 2) tensor. It should be clear that we can extend the definition to the direct
product of any number M of 1-form factors to produce a (0, M) tensor and so forth.
Recall that we discussed in Sect. 4.3 a basis for 1-forms which we denoted as ω
α .
We can similarly show that there exists a basis for the product of two 1-forms or (0,
2) tensors. Indeed the basis is a linear combination of the direct product of the ω
α .
We write that linear combination as
f = f αβ ω
α
⊗ ω
β
.
(4.57)