42
4 Riemann Spaces and Tensors
This is a common situation in mathematics; for example relations in group theory
may be represented in terms of matrices and row and column vectors. Probably the
most important example is in quantum physics: the wave function may be viewed
as an inner product of a Hilbert space state vector with a position eigenstate, or as a
representation of the state.
Digression 4.1 Let is digress briefly to ask an important question: what
happens if we use different coordinates and thus different basis vectors? The
displacement d s represents a real physical distance, independent of the coordinate system, so it should not change, but its components change. Thus we
may write for two systems, unprimed and primed,
d s = =
e j dx
j
= =
e j
∂ x
j
∂ x k
dx
k
= =
e
k dx
k
.
(4.26)
Since the coordinate displacements are arbitrary we must have the following
transformation rule for the basis vectors and the differentials,
e
k =
∂ x
j
∂ x k
e j , dx
i
=
∂ x
i
∂ x k
dx
k
,
(4.27)
If we compare these two expressions we see that the coordinate differentials and
the basis vectors transform in an opposite sense, what is called contragrediently.
Objects which transform in the same sense are said to transform cogradiently.
Our next mathematical object is a 1-form: 1-forms comprise a dual space to
vectors. They are defined to operate linearly on vectors to give real scalars; a 1-form
˜
p operates on a vector
V and maps it into a scalar according to the defining rule
˜
p
V
= ˜
p
e j V
j
= V
j
˜
p
e j
= V
j p j , p j = ˜
p
e j
.
(4.28)
This defines the components p j of the form. Another notation often used is equivalent
to (4.28), but emphasizes the symmetry between the vector space and the dual 1-form
space,
˜
p(
V ) = = ˜
p,
V = V
j p j .
(4.29)
This idea is of course familiar in elementary matrix theory, where row vectors form a
dual space to column vectors, and map them into single numbers; the column vectors
could also be thought of as mapping row vectors into scalars.
Being in a linear vector space the 1-forms will have a basis, which we denote
˜
ω
m . We assume the expansion coefficients are the components defined in (4.28), so
˜
p = p m ˜
ω
m . Then we see that the basis 1-forms and basis vectors must obey an
4 Riemann Spaces and Tensors
This is a common situation in mathematics; for example relations in group theory
may be represented in terms of matrices and row and column vectors. Probably the
most important example is in quantum physics: the wave function may be viewed
as an inner product of a Hilbert space state vector with a position eigenstate, or as a
representation of the state.
Digression 4.1 Let is digress briefly to ask an important question: what
happens if we use different coordinates and thus different basis vectors? The
displacement d s represents a real physical distance, independent of the coordinate system, so it should not change, but its components change. Thus we
may write for two systems, unprimed and primed,
d s = =
e j dx
j
= =
e j
∂ x
j
∂ x k
dx
k
= =
e
k dx
k
.
(4.26)
Since the coordinate displacements are arbitrary we must have the following
transformation rule for the basis vectors and the differentials,
e
k =
∂ x
j
∂ x k
e j , dx
i
=
∂ x
i
∂ x k
dx
k
,
(4.27)
If we compare these two expressions we see that the coordinate differentials and
the basis vectors transform in an opposite sense, what is called contragrediently.
Objects which transform in the same sense are said to transform cogradiently.
Our next mathematical object is a 1-form: 1-forms comprise a dual space to
vectors. They are defined to operate linearly on vectors to give real scalars; a 1-form
˜
p operates on a vector
V and maps it into a scalar according to the defining rule
˜
p
V
= ˜
p
e j V
j
= V
j
˜
p
e j
= V
j p j , p j = ˜
p
e j
.
(4.28)
This defines the components p j of the form. Another notation often used is equivalent
to (4.28), but emphasizes the symmetry between the vector space and the dual 1-form
space,
˜
p(
V ) = = ˜
p,
V = V
j p j .
(4.29)
This idea is of course familiar in elementary matrix theory, where row vectors form a
dual space to column vectors, and map them into single numbers; the column vectors
could also be thought of as mapping row vectors into scalars.
Being in a linear vector space the 1-forms will have a basis, which we denote
˜
ω
m . We assume the expansion coefficients are the components defined in (4.28), so
˜
p = p m ˜
ω
m . Then we see that the basis 1-forms and basis vectors must obey an