40
4 Riemann Spaces and Tensors
Theorem 2 The inner product of a contravariantand covariant vector, defined as
V
β W β , is a scalar. We use the transformation of the vectors and Theorem 1 to
calculate the inner product in the barred system to see that it is an invariant,
V
β W β =
∂ x
β
∂ x η
∂ x
σ
∂ x
β
V
η W σ = δ
σ
η V
η W σ = V
η W η .
(4.19)
Thus we see that we may think of covariant vectors as objects that map contravariant
vectors into scalars. We will return to this idea when we discuss forms in the next
section.
4.3 Vectors and 1-Forms, Abstract View
We can connect the above idea of vectors as component n-tuples with the idea of
intrinsic or abstract vectors, often represented in physics by arrows, and in the process
introduce a definition of a metric. We may think of such vectors as intrinsic or abstract,
but the word physical is also appropriate since they are taken to exist independently of
the coordinate system and are invariant. These abstract vectors are taken to exist in an
idealized physical world, whereas component vectors only exist when we represent
them in terms of a specific coordinate system.
Look at a single point P in a Reimann space. We introduce a set of basis vectors
e k
along the grid lines illustrated in Fig. 4.5 for two dimensions, but we do not assume
the basis is orthonormal. The basis set spans a vector space associated with that point.
Such a basis is naturally called a coordinate basis.
A small displacement d s along a curve or in some specified direction is given by
d s = =
e j dx
j
,
(4.20)
and its square is given by
d s
2
= ds
2
=
e j dx
j
·
e k dx
k
=
e j · ·
e k
dx
j dx
k
= g jk dx
j dx
k
,
g jk = =
e j · ·
e k .
(4.21)
Fig. 4.5 A coordinate basis in two dimensions, with a small displacement vector d s
4 Riemann Spaces and Tensors
Theorem 2 The inner product of a contravariantand covariant vector, defined as
V
β W β , is a scalar. We use the transformation of the vectors and Theorem 1 to
calculate the inner product in the barred system to see that it is an invariant,
V
β W β =
∂ x
β
∂ x η
∂ x
σ
∂ x
β
V
η W σ = δ
σ
η V
η W σ = V
η W η .
(4.19)
Thus we see that we may think of covariant vectors as objects that map contravariant
vectors into scalars. We will return to this idea when we discuss forms in the next
section.
4.3 Vectors and 1-Forms, Abstract View
We can connect the above idea of vectors as component n-tuples with the idea of
intrinsic or abstract vectors, often represented in physics by arrows, and in the process
introduce a definition of a metric. We may think of such vectors as intrinsic or abstract,
but the word physical is also appropriate since they are taken to exist independently of
the coordinate system and are invariant. These abstract vectors are taken to exist in an
idealized physical world, whereas component vectors only exist when we represent
them in terms of a specific coordinate system.
Look at a single point P in a Reimann space. We introduce a set of basis vectors
e k
along the grid lines illustrated in Fig. 4.5 for two dimensions, but we do not assume
the basis is orthonormal. The basis set spans a vector space associated with that point.
Such a basis is naturally called a coordinate basis.
A small displacement d s along a curve or in some specified direction is given by
d s = =
e j dx
j
,
(4.20)
and its square is given by
d s
2
= ds
2
=
e j dx
j
·
e k dx
k
=
e j · ·
e k
dx
j dx
k
= g jk dx
j dx
k
,
g jk = =
e j · ·
e k .
(4.21)
Fig. 4.5 A coordinate basis in two dimensions, with a small displacement vector d s