Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 6; Problem 6.1.
The inverse metric tensor is defined so that
If the metric tensor is viewed as a matrix, then this is equivalent to saying . The transformation property of can be worked out by direct calculation, using the transformation of and the fact that is invariant.
We can try the transformation
Substituting, we get
On line 2 we used and on line 4 we used . Thus is a rank-2 contravariant tensor, and is the inverse of which is a rank-2 covariant tensor. Since the matrix inverse is unique (basic fact from matrix algebra), we can use the standard techniques of matrix algebra to calculate the inverse.
In rectangular coordinates, since the metric is diagonal with all diagonal elements equal to 1. In polar coordinates in 2-d,
so the inverse is
A contravariant vector can be lowered (converted to a covariant vector) by multiplying by :
The covariant vector can be converted back into a contravariant vector by raising its index:
If we start with a vector in rectangular coordinates, we can convert it to polar coordinates:
We can lower these components by multiplying by
The square magnitude is
(No implied sum on the RHS in line 1.)