General coordinate transformations and tensors

TENSORS
Thus, by inverting the matrix G in (26.60), we find that the elements g ij are given in
cylindrical polar coordinates by


1
0
0
ij
2

0 1/ρ 0  . Ĝ = [g ] =
0
0
1
So far we have not considered the components of the metric tensor gji with one
subscript and one superscript. By analogy with (26.56), these mixed components
are given by
gji = ei · ej = δij ,
and so the components of gji are identical to those of δji . We may therefore
consider the δji to be the mixed components of the metric tensor g.
26.16 General coordinate transformations and tensors
We now discuss the concept of general transformations from one coordinate
system, u1 , u2 , u3 , to another, u 1 , u 2 , u 3 . We can describe the coordinate transform
using the three equations
u = u (u1 , u2 , u3 ),
i
i
for i = 1, 2, 3, in which the new coordinates u i can be arbitrary functions of the old
ones ui rather than just represent linear orthogonal transformations (rotations)
of the coordinate axes. We shall assume also that the transformation can be
inverted, so that we can write the old coordinates in terms of the new ones as
ui = ui (u , u , u ),
1
2
3
As an example, we may consider the transformation from spherical polar to
Cartesian coordinates, given by
x = r sin θ cos φ,
y = r sin θ sin φ,
z = r cos θ,
which is clearly not a linear transformation.
The two sets of basis vectors in the new coordinate system, u1 , u2 , u3 , are given
as in (26.55) by
ei =
∂r
∂u i
and
e = ∇u .
i
i
Considering the first set, we have from the chain rule that
∂u i ∂r
∂r
=
,
∂u j
∂u j ∂u i
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(26.66)
26.16 GENERAL COORDINATE TRANSFORMATIONS AND TENSORS
so that the basis vectors in the old and new coordinate systems are related by
ej =
∂u i e.
∂u j i
(26.67)
Now, since we can write any arbitrary vector a in terms of either basis as
a = a ei = a j ej = a j
i
∂u i e,
∂u j i
it follows that the contravariant components of a vector must transform as
a =
i
∂u i j
a .
∂u j
(26.68)
In fact, we use this relation as the defining property for a set of quantities ai to
form the contravariant components of a vector.
Find an expression analogous to (26.67) relating the basis vectors ei and e i in the two
coordinate systems. Hence deduce the way in which the covariant components of a vector
change under a coordinate transformation.
If we consider the second set of basis vectors in (26.66), e i = ∇u i , we have from the chain
rule that
∂u j
∂u j ∂u i
= i
∂x
∂u ∂x
and similarly for ∂u j /∂y and ∂u j /∂z. So the basis vectors in the old and new coordinate
systems are related by
ej =
∂u j i
e .
∂u i
(26.69)
For any arbitrary vector a,
∂u j i
e
∂u i
and so the covariant components of a vector must transform as
a = ai e = aj e j = aj
i
∂u j
aj .
(26.70)
∂u i
Analogously to the contravariant case (26.68), we take this result as the defining property
of the covariant components of a vector. ai =
We may compare the transformation laws (26.68) and (26.70) with those for
a first-order Cartesian tensor under a rigid rotation of axes. Let us consider
a rotation of Cartesian axes xi through an angle θ about the 3-axis to a new
set x i , i = 1, 2, 3, as given by (26.7) and the inverse transformation (26.8). It is
straightforward to show that
∂x i
∂x j
=
= Lij ,
i
∂x j
∂x
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TENSORS
where the elements Lij are given by

cos θ
L =  − sin θ
0

0
0 .
1
sin θ
cos θ
0
Thus (26.68) and (26.70) agree with our earlier definition in the special case of a
rigid rotation of Cartesian axes.
Following on from (26.68) and (26.70), we proceed in a similar way to define general tensors of higher rank. For example, the contravariant, mixed and
covariant components, respectively, of a second-order tensor must transform as
follows:
∂u i ∂u j kl
T ;
∂uk ∂ul
∂u i ∂ul
= k j T kl ;
∂u ∂u
∂uk ∂ul
= i j Tkl .
∂u ∂u
contravariant components,
T =
mixed components,
T j
covariant components,
T ij
ij
i
It is important to remember that these quantities form the components of the
same tensor T but refer to different tensor bases made up from the basis vectors
of the different coordinate systems. For example, in terms of the contravariant
components we may write
T = T ij ei ⊗ ej = T ei ⊗ ej .
ij
We can clearly go on to define tensors of higher order, with arbitrary numbers
of covariant (subscript) and contravariant (superscript) indices, by demanding
that their components transform as follows:
T
ij···k
lm···n
=
∂u i ∂u j
∂u k ∂ud ∂ue
∂uf
· · · c l m · · · n T ab···cde···f .
a
b
∂u ∂u
∂u ∂u ∂u
∂u
(26.71)
Using the revised summation convention described in section 26.14, the algebra
of general tensors is completely analogous to that of the Cartesian tensors
discussed earlier. For example, as with Cartesian coordinates, the Kronecker
delta is a tensor provided it is written as the mixed tensor δji since
δj =
i
∂u i ∂ul k
∂u i ∂uk
∂u i
δl = k j = j = δji ,
j
k
∂u ∂u
∂u ∂u
∂u
where we have used the chain rule to justify the third equality. This also shows
that δji is isotropic. As discussed at the end of section 26.15, the δji can be
considered as the mixed components of the metric tensor g.
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