Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 6; Problem 6.11.
In basic physics, the moment of inertia is usually defined by the integral
where is the perpendicular distance of the volume element from the axis of rotation and is the mass density. In fact, this is true only for special cases, where the mass distribution is symmetric with respect to the axis (cases such as a sphere, cylinder, etc). In rotational motion, the angular momentum of a spinning rigid body (that is, a body which does not deform as it moves) is given by where is the angular velocity vector.
In the more general case, where the object isn’t symmetric, the moment of inertia becomes a tensor, and the angular momentum equation becomes (in 3-d):
The derivation of this tensor would take us too far afield here, but basically, the off-diagonal elements of measure the amount of asymmetry in various directions. For example, in rectangular coordinates
where the coordinates are measured relative to the centre of mass. For a symmetric object, this integral is always zero, since any mass element at is balanced by an equal mass element at .
If we start with a standard 3-d rectangular system and rotate it by an angle about the axis to get the primed system, then the transformation can be found by considering the projection of the unit vectors and onto the and axes.
By taking the scalar product of both these equations with we can get the components of along the original axes, and similarly for . For example is the component of in the direction. We get
If we now take an arbitrary vector and take its scalar product with , we get
Similarly for :
The transformation partials are then
Now suppose we have an inertia tensor that is diagonal (this is true for symmetric objects, but it is always possible to find some principal axes where this is true for any object) so that
We can transform this to the rotated system using the usual tensor transformation rule:
This is equivalent to a matrix product as we can see by taking it in stages. First consider the intermediate product
This is the sum over the product of elements in the th row of with elements in the th column of . Thus this is a matrix product with the factors in the order . Now consider
This time, we’re summing over the product of elements in the th row of with elements in the th row of . In order to make this a matrix product, we have to sum over the row elements of one matrix multiplied by the column elements of the second matrix, so we need to take the transpose of to make this work. That is