Tensors for Physics

134 8 Integration of Fields

8.3.3 Application: Moment of Inertia Tensor

The moment of inertia tensorΘμν, introduced in Sect.4.3.2, can also be expressed

as a volume integral over the mass densityρof a solid body, viz.:

Θμν=

∫

V

ρ(r)(r^2 δμν−rμrν)d^3 r. (8.65)

The moment of inertia tensor, evaluated either as a sum over discrete masses or

a volume integral, depends on the choice of the origin. When one talks aboutthe

moment of inertia tensor of a mass distribution, it is understood thatr=0, in (8.65),

corresponds to its center of mass. In the general case, the effective moment of inertia

tensor, entering the linear relation between the angular momentum and the angular

velocity, is

Θμνeff=M(R^2 δμν−RμRν)+Θcmμν, (8.66)

whereMis the total mass,Ris the position of the center of mass, and it is understood

thatΘμνcmis for a rotation around the center of mass. Relation (8.66) is referred to

as∫ Steiner’s law. It can be derived from (8.65) withr=R+r′. For the proof, use

Vρr

′d (^3) r′=0. When there is no danger of confusionΘcm
μνwill be denoted byΘμν,
in the following.
As also pointed out in Sect.4.3.2,themoment of inertiafor a rotation about a
fixed axis is defined via the linear relation between the component of the angular
momentum parallel to this axis and the magnitudewof the angular velocity. With
the unit vector̂wμ, parallel to the axis of rotation, the moment of inertia isΘ=
̂wμΘμν̂wν, and consequently

Θ=

∫

V

ρ(r)r⊥^2 d^3 r, (8.67)

where

r^2 ⊥=r^2 −̂wμrμ̂wνrν=r^2 −(̂w·r)^2

is the square of the shortest distance of a mass element atr, from the rotation axis.

The origin of the position vector is a point on the rotation axis. With thez-axis chosen

parallel to the rotation axis,r⊥^2 is justr^2 −z^2 =x^2 +y^2.

By definition, the moment of inertia tensor is symmetric and positive definite.

In general, it has three eigenvaluesΘ(^1 ),Θ(^2 )andΘ(^3 ), which are the moments of

inertia for rotations about the three principal axes. An object with three different

principal moments of inertia is referred to as asymmetric top. A symmetric top has

two equal eigenvalues, e.g.Θ^1 =Θ(^2 )=Θ(^3 ), for the spherical top, all three are

equal. These different types of symmetry of the moment of inertia tensor result from