4.3 Vectors and 1-Forms, Abstract View
41
The metric g jk defined in this way, as the inner product of basis vectors, agrees
with the line element expression (4.4) and implies that the metric is intrinsically
symmetric. Note that, for example, a coordinate interval dx
1 along the first axis
corresponds to an invariant physical distance
ds =
√ g 11 dx
1
,
(4.22)
Here g 11 is the square of
e j and could be anything we choose; we assume in this
example that g 11 is positive. This shows clearly the role of the metric in relating
coordinate distances to physical distances: only the combination of coordinates and
metric has physical meaning. Note in particular that the dimensions of the coordinates
need not be distances; for example they could be angles as in polar or spherical
coordinates; but the dimension of the metric components must be such that the
product in (4.22) is a distance.
Now consider any vector
V at P. We expand it in the coordinate basis as we did
for the small displacement vector in Fig. 4.5, and calculate its square
V = =
e j V
j
,
V
2
=
e j V
j
·
e k V
k
=
e j · ·
e k
V
j V
k
= g jk V
j V
k
.
(4.23)
Here V
k are the vector components with respect to the coordinate basis; they
correspond to the contravariant component vector in (4.14).
The vector
V can also be characterized in terms of its projections on the basis
vectors; denoting these projections with lower indices we define explicitly,
V i =
V · ·
e i =
e j V
j
· ·
e i =
e j · ·
e i
V
j
= g i j V
j
.
(4.24)
The V i correspond to the covariant component vector in (4.16). Just as in special
relativity the metric lowers the index position according to (4.24). Conversely, we
may define the inverse g
jn of the metric as the inverse of its associated matrix and
invert the relation (4.24). This gives
g
jn g nk = δ
j
k , V
j
= g
jk V k .
(4.25)
Thus we are led to the idea of lowering and raising indices with the metric just as in
the previous Sect. 4.2. It is a natural extension of the ideas and notation of special
relativity in Part I.
An important basic mathematical point which we emphasize here is that the real
n-tuples V
i introduced with respect to a coordinate basis may be viewed in two
separate ways:
1. As components of a vector with respect to a basis, as in (4.23).
2. As a representation of the vector.
41
The metric g jk defined in this way, as the inner product of basis vectors, agrees
with the line element expression (4.4) and implies that the metric is intrinsically
symmetric. Note that, for example, a coordinate interval dx
1 along the first axis
corresponds to an invariant physical distance
ds =
√ g 11 dx
1
,
(4.22)
Here g 11 is the square of
e j and could be anything we choose; we assume in this
example that g 11 is positive. This shows clearly the role of the metric in relating
coordinate distances to physical distances: only the combination of coordinates and
metric has physical meaning. Note in particular that the dimensions of the coordinates
need not be distances; for example they could be angles as in polar or spherical
coordinates; but the dimension of the metric components must be such that the
product in (4.22) is a distance.
Now consider any vector
V at P. We expand it in the coordinate basis as we did
for the small displacement vector in Fig. 4.5, and calculate its square
V = =
e j V
j
,
V
2
=
e j V
j
·
e k V
k
=
e j · ·
e k
V
j V
k
= g jk V
j V
k
.
(4.23)
Here V
k are the vector components with respect to the coordinate basis; they
correspond to the contravariant component vector in (4.14).
The vector
V can also be characterized in terms of its projections on the basis
vectors; denoting these projections with lower indices we define explicitly,
V i =
V · ·
e i =
e j V
j
· ·
e i =
e j · ·
e i
V
j
= g i j V
j
.
(4.24)
The V i correspond to the covariant component vector in (4.16). Just as in special
relativity the metric lowers the index position according to (4.24). Conversely, we
may define the inverse g
jn of the metric as the inverse of its associated matrix and
invert the relation (4.24). This gives
g
jn g nk = δ
j
k , V
j
= g
jk V k .
(4.25)
Thus we are led to the idea of lowering and raising indices with the metric just as in
the previous Sect. 4.2. It is a natural extension of the ideas and notation of special
relativity in Part I.
An important basic mathematical point which we emphasize here is that the real
n-tuples V
i introduced with respect to a coordinate basis may be viewed in two
separate ways:
1. As components of a vector with respect to a basis, as in (4.23).
2. As a representation of the vector.