TENSORS
uito anotheru′i, we may define the Jacobian of the transformation (see chapter 6)
as the determinant of the transformation matrix [∂u′i/∂uj]: this is usually denoted
by
J=∣
∣
∣
∣∂u′
∂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 JacobianJ, we define a relative tensor of weightwas one whosecomponents transform as follows:
T′ij···k
lm···n=∂u′i
∂ua∂u′j
∂ub···∂u′k
∂uc∂ud
∂u′l∂ue
∂u′m···∂uf
∂u′nTab···cde···f∣
∣
∣
∣∂u
∂u′∣
∣
∣
∣w
.
(26.74)Comparing this expression with (26.71), we see that a true (orabsolute) general
tensor may be considered as a relative tensor of weightw=0.Ifw=−1, on the
other hand, the relative tensor is known as a generalpseudotensorand ifw=1
as atensor density.
It is worth comparing (26.74) with the definition (26.39) of a Cartesian pseu-dotensor. For the latter, we are concerned only with its behaviour under a rotation
(proper or improper) of Cartesian axes, for which the JacobianJ=±1. Thus,
general relative tensors of weightw=−1andw= 1 would both satisfy the
definition (26.39) of a Cartesian pseudotensor.
If thegijare the covariant components of the metric tensor, show that the determinantg
of the matrix[gij]is a relative scalar of weightw=2.The componentsgijtransform as
gij′=∂uk
∂u′i∂ul
∂u′jgkl.Defining the matricesU=[∂ui/∂u′j],G=[gij]andG′=[g′ij], we may write this expression
as
G′=UTGU.Taking the determinant of both sides, we obtain
g′=|U|^2 g=∣
∣
∣∣∂u
∂u′∣
∣
∣∣
2
g,which shows thatgis a relative scalar of weightw=2.
From the discussion in section 26.8, it can be seen thatijkis a covariantrelative tensor of weight−1. We may also define the contravariant tensorijk,
which is numerically equal toijkbut is a relative tensor of weight +1.
If two relative tensors have weightsw 1 andw 2 respectively then, from (26.74),