4.5 Tensors, Abstract View
49
To see that this is indeed a basis we verify that it produces the same result (4.54)
when operating on two vectors by writing out its operation
f
V ,
W
= f αβ ω
α
⊗ ω
β
V ,
W
= f αβ ω
α
V
ω
β
W
= f αβ V
α W
β
. (4.58)
It should be clear that we can extend this idea to any number of factors in the direct
product and thereby have a basis for (0, M) tensors.
Recall that the coordinate basis may be written in terms of the gradients of the
coordinates as in (4.31). This allows us to write a curious and useful expression for
the metric tensor from (4.58),
g = g αβ dx
α
⊗ dx
β
.
(4.59)
This looks like the expression for the line element but is a relation between forms.
From the above definitions and (4.59) a (0, 2) we see that a tensor, such as the
metric, can also be viewed as producing a 1-form from a vector if we leave the second
space blank, or g
V , −
; this maps vectors to scalars according to (4.59) as follows
g
V , −
= g αβ ˜
dx
α
⊗ ˜
dx
β
V , −
= g αβ ˜
dx
α
V
˜
dx
β
= g αβ V
α ˜
dx
β
= V β ˜
dx
β
.
(4.60)
That is, the metric lowers the index to produce the components of the 1-form.
Finally, it is now rather obvious how to define a tensor in general in terms of the
coordinate basis vectors and basis forms: an (M, N) tensor is a linear mapping of N
vectors and M 1-forms to the scalars; it may be expanded in terms of the bases and
components as
T = T
α... β... ( e α ⊗ . . .)( ˜
dx
β
⊗ . . .),
⊗ = direct product.
(4.61)
The direct product of the basis vectors in the above is the obvious analog of the
direct product of the basis 1-forms. The relation between the abstract tensor and its
components in terms of the coordinate basis vectors and 1-forms is thus fundamental
and clear.
4.6 Tetrads and n-Trads
In general relativity we often find it useful to use tetrads, a set of four basis vectors
that forms an orthonormal basis as in special relativity. This sets up a reference frame
at a point that is analogous to the reference frame of special relativity. The tetrad
49
To see that this is indeed a basis we verify that it produces the same result (4.54)
when operating on two vectors by writing out its operation
f
V ,
W
= f αβ ω
α
⊗ ω
β
V ,
W
= f αβ ω
α
V
ω
β
W
= f αβ V
α W
β
. (4.58)
It should be clear that we can extend this idea to any number of factors in the direct
product and thereby have a basis for (0, M) tensors.
Recall that the coordinate basis may be written in terms of the gradients of the
coordinates as in (4.31). This allows us to write a curious and useful expression for
the metric tensor from (4.58),
g = g αβ dx
α
⊗ dx
β
.
(4.59)
This looks like the expression for the line element but is a relation between forms.
From the above definitions and (4.59) a (0, 2) we see that a tensor, such as the
metric, can also be viewed as producing a 1-form from a vector if we leave the second
space blank, or g
V , −
; this maps vectors to scalars according to (4.59) as follows
g
V , −
= g αβ ˜
dx
α
⊗ ˜
dx
β
V , −
= g αβ ˜
dx
α
V
˜
dx
β
= g αβ V
α ˜
dx
β
= V β ˜
dx
β
.
(4.60)
That is, the metric lowers the index to produce the components of the 1-form.
Finally, it is now rather obvious how to define a tensor in general in terms of the
coordinate basis vectors and basis forms: an (M, N) tensor is a linear mapping of N
vectors and M 1-forms to the scalars; it may be expanded in terms of the bases and
components as
T = T
α... β... ( e α ⊗ . . .)( ˜
dx
β
⊗ . . .),
⊗ = direct product.
(4.61)
The direct product of the basis vectors in the above is the obvious analog of the
direct product of the basis 1-forms. The relation between the abstract tensor and its
components in terms of the coordinate basis vectors and 1-forms is thus fundamental
and clear.
4.6 Tetrads and n-Trads
In general relativity we often find it useful to use tetrads, a set of four basis vectors
that forms an orthonormal basis as in special relativity. This sets up a reference frame
at a point that is analogous to the reference frame of special relativity. The tetrad