Inertial tensor remains diagonal during a shift along a principle axis

[Happiness]

In the middle of the below paragraph: "only if the shift vector $R$ is along one of the principal axes relative to the center of mass will the difference tensor be diagonal in that system." I suppose the difference tensor means new inertial tensor $-$ old inertial tensor.

That means the new inertial tensor is also diagonal. Suppose we let the principal axis along which the shift happens be the $x$ axis. That means $x_i$ changes but $I_{xy}$ and $I_{xz}$ remain $0$. By (5.7), that means that $m_iy_i=0$ and $m_iz_i=0$ for all values of $x$, since the shift could be by any arbitrary amount in the $x$ direction.

However, this intuitively does not seem to be true in general for an arbitrarily shaped object.

Consider an object with a uniform cross section (drawn below). $m_iy_i=0$ (for all values of $x$) means that for all points on line 1, $\Sigma_i h_i=\Sigma_i d_i$, and $m_ix_i=0$ (for all values of $y$ [now we consider the shift along another principle axis, the $y$ axis]) means that for all points on line 2, $\Sigma_i l_i=\Sigma_i r_i$. For these to be true, we must have tangent 1a and tangent 1b to be parallel to line 2, and also tangent 2a and tangent 2b to be parallel to line 1. But these seem unachievable in general for an arbitrarily shaped cross sections.

 

[gtoguy97]

Therefore, only if the shift vector $R$ is along one of the principal axes relative to the center of mass will the difference tensor be diagonal in that system.