Relative tensors

26.17 RELATIVE TENSORS
Show that the quantities gij = ei · ej form the covariant components of a second-order
tensor.
In the new (primed) coordinate system we have
gij = ei · ej ,
but using (26.67) for the inverse transformation, we have
ei =
∂uk
ek ,
∂u i
and similarly for ej . Thus we may write
gij =
∂uk ∂ul
∂uk ∂ul
ek · el = i j gkl ,
∂u i ∂u j
∂u ∂u
which shows that the gij are indeed the covariant components of a second-order tensor
(the metric tensor g). A similar argument to that used in the above example shows that the quantities
g ij form the contravariant components of a second-order tensor which transforms
according to
g =
ij
∂u i ∂u j kl
g .
∂uk ∂ul
In the previous section we discussed the use of the components gij and g ij in
the raising and lowering of indices in contravariant and covariant vectors. This
can be extended to tensors of arbitrary rank. In general, contraction of a tensor
with gij will convert the contracted index from being contravariant (superscript)
to covariant (subscript), i.e. it is lowered. This can be repeated for as many indices
are required. For example,
Tij = gik T k j = gik gjl T kl .
(26.72)
Similarly contraction with g ij raises an index, i.e.
T ij = g ik Tkj = g ik g jl Tkl .
(26.73)
That (26.72) and (26.73) are mutually consistent may be shown by using the fact
that g ik gkj = δji .
26.17 Relative tensors
In section 26.10 we introduced the concept of pseudotensors in the context of the
rotation (proper or improper) of a set of Cartesian axes. Generalising to arbitrary
coordinate transformations leads to the notion of a relative tensor.
For an arbitrary coordinate transformation from one general coordinate system
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TENSORS
ui to another u i , we may define the Jacobian of the transformation (see chapter 6)
as the determinant of the transformation matrix [∂u i /∂u j ]: this is usually denoted
by
∂u .
J = ∂u Alternatively, we may interchange the primed and unprimed coordinates to
obtain |∂u/∂u | = 1/J: unfortunately this also is often called the Jacobian of the
transformation.
Using the Jacobian J, we define a relative tensor of weight w as one whose
components transform as follows:
∂u w
∂u i ∂u j
∂u k ∂ud ∂ue
∂uf ab···c
ij···k
.
=
·
·
·
·
·
·
T
T
lm···n
de···f ∂ua ∂ub
∂uc ∂u l ∂u m
∂u n
∂u (26.74)
Comparing this expression with (26.71), we see that a true (or absolute) general
tensor may be considered as a relative tensor of weight w = 0. If w = −1, on the
other hand, the relative tensor is known as a general pseudotensor and if w = 1
as a tensor density.
It is worth comparing (26.74) with the definition (26.39) of a Cartesian pseudotensor. For the latter, we are concerned only with its behaviour under a rotation
(proper or improper) of Cartesian axes, for which the Jacobian J = ±1. Thus,
general relative tensors of weight w = −1 and w = 1 would both satisfy the
definition (26.39) of a Cartesian pseudotensor.
If the gij are the covariant components of the metric tensor, show that the determinant g
of the matrix [gij ] is a relative scalar of weight w = 2.
The components gij transform as
gij =
∂uk ∂ul
gkl .
∂u i ∂u j
Defining the matrices U = [∂ui /∂u j ], G = [gij ] and G = [gij ], we may write this expression
as
G = UT GU.
Taking the determinant of both sides, we obtain
∂u 2
g = |U|2 g = g,
∂u
which shows that g is a relative scalar of weight w = 2. From the discussion in section 26.8, it can be seen that ijk is a covariant
relative tensor of weight −1. We may also define the contravariant tensor ijk ,
which is numerically equal to ijk but is a relative tensor of weight +1.
If two relative tensors have weights w1 and w2 respectively then, from (26.74),
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