9.4.3 Physical components of tensors

In many applications, it is useful to introduce the physical components of a tensor (see Section 7.4 of Aris or 4.8 of Weinberg). For example, the velocity field using spherical coordinates is often written

 

$$\left( \sqrt{g_{\lambda \lambda}} \, u^{\lambda}, \sqrt{g_{\phi \phi}} \, u^{\phi}, \sqrt{g_{rr}} \, u^{r} \right)$$ (u, v, w) =

$$\left( r \, \cos\phi \, \frac{D \lambda}{Dt}, r \, \frac{D\phi}{Dt}, \frac{Dr}{Dt} \right).$$ = (9.95)

Additionally, the infinitesimal displacements along the coordinate directions on the sphere are given by

 

$$(\delta x, \delta y, \delta z) \left( (r \, \cos\phi) \, \delta \lambda, r \, \delta \phi, \delta r \right).$$ = (9.96)

More generally, for any orthogonal coordinate system, the physical components of the displacement will be written

$$(\delta x, \delta y, \delta z) (h_{1} \, \delta \xi^{1}, \, h_{2} \, \delta \xi^{2}, \, h_{3} \, \delta \xi^{3})$$ = (9.97)

Likewise, the physical components of the velocity are written

$$(h_{1} \, u^{1}, \, h_{2} \, u^{2}, \, h_{3} \, u^{3}).$$ (u,v,w) = (9.98)

As such, for example,

$$u^{1}_{,1} = \sqrt{g_{11}} \, (u/\sqrt{g_{11}})_{,x} = h_{1} \, (u/h_{1})_{,x}$$ (9.99)

Note that the traditional Cartesian notation x,y,z is used for convenience; the coordinates are generally curvilinear. The key property of the physical components of a tensor is that each has the same dimensions; e.g., length for the physical displacement components, length/time for the physical velocity components, etc. Importantly, the physical components are not components to a true tensor since the tensorial transformation rules are corrupted by the square root of the metric. Correspondingly, the physical components of the partial derivative operator do not necessarily commute; i.e., $\partial_{x} \, \partial_{y} = h_{1}^{-1} \partial_{1} \, h_{2}^{-1} \partial_{2}$ equals $\partial_{y} \, \partial_{x}$ only for a constant metric. Hence, it is best to perform mathematical manipulations with the tensor quantities, and only after establishing the final result should the physical components be introduced before discretizing. This is the approach taken in the following.