Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.17 Relative tensors

Show that the quantitiesgij=ei·ejform the covariant components of a second-order

tensor.

In the new (primed) coordinate system we have

g′ij=e′i·e′j,

but using (26.67) for the inverse transformation, we have

e′i=

∂uk

∂u′i

ek,

and similarly fore′j.Thuswemaywrite

gij′=

∂uk

∂u′i

∂ul

∂u′j

ek·el=

∂uk

∂u′i

∂ul

∂u′j

gkl,

which shows that thegijare indeed the covariant components of a second-order tensor
(themetrictensorg).

A similar argument to that used in the above example shows that the quantities

gijform the contravariant components of a second-order tensor which transforms

according to

g′ij=

∂u′

i

∂uk

∂u′

j

∂ul

gkl.

In the previous section we discussed the use of the componentsgijandgijin

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

withgijwill 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=gikTkj=gikgjlTkl. (26.72)

Similarly contraction withgijraises an index, i.e.

Tij=gikTkj=gikgjlTkl. (26.73)

That (26.72) and (26.73) are mutually consistent may be shown by using the fact

thatgikgkj=δij.

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 arelative tensor.

For an arbitrary coordinate transformation from one general coordinate system