Mathematical Tools for Physics

12—Tensors 378

In short, the matrix(bki)is the inverse transpose of the matrix(akj). Recall, I’m using the convention that the
first index represents the row of the matrix and the second index the column, whether the index is up or down.
If you were to make the restriction to orthonormal bases then not only would the reciprocal and direct bases
be the same, the distinction between upper and lower indices would vanish, and necessarily all changes of basis
would be orthogonal transformations (rotations or reflections). In this case the above matrices(a)and(b)are
equal since an orthogonal matrix is equal to its inverse transpose.

Raising and Lowering
There is a particular basis change of general use. This is the change from direct to reciprocal basis or vice-versa.
It’s perfectly legitimate to look on this as a change of basis because these two sets of basis vectors both span the
space and so one must be expressible in terms of the other. The transformation is accomplished by means of the
components of the metric tensor:
~ei=gji~ej

You can prove this relation by multiplying both sides by~ek.

~ek.~ei=~ek.gji~ej=gjiδkj=gki

but this is just the definition of the components ofg. Similarly,

~ei=gji~ej

and the proof is the same, multiply by~ekto get

~ek.~ei=~ek.gji~ej=gjiδkj=gki

This operation also changes covariant components to contravariant and the reverse. It isn’t restricted to
vector components, but works for any rank tensor.

vi=gijvj, Tij=gikgj`Tk`, Tij=gikTkj, etc. (37)

These operations are called raising and lowering as you might expect, and one is the inverse of the other (consistent
with the matrix properties found in the special case of Eq. ( 33 )). The sort of manipulation that it represents is
one of the more common tools in manipulating tensors.