Interpretation of tensor indices

[Niles]

Hi

I am trying to figure out the physical interpretation of tensor indices. E.g., if we have an anisotropic material, then the tensor element of the susceptibility x_x,y gives the response in the x-direction of an electric field in the y-direction.

Lets not go up one dimension and look at e.g. x_x,y,z. This I guess gives the response in the x-direction of an electric field in the y- and z-direction?


Niles.

 

[tiny-tim]

Hi Niles!

I am trying to figure out the physical interpretation of tensor indices.

It depends what you apply them to.

Susceptibility has one input vector and one output vector.

A 3-index tensor could have two different input vectors and one output vector, or one input vector used twice, and one output vector, or vice versa.

For a 4-index example, take the geodesic deviation equation:

$$\frac{D^2\,\delta x^{\alpha}}{D\tau^2}\ =\ -\,R^{\alpha}_{\ \mu\beta\sigma}\,V^{\mu}\,V^{\sigma}\,\delta x^{\beta}$$

Here, the 4-index tensor R acts on x, and on V twice (that's two input vectors of which one is used twice, and one output vector), but an identical tensor could conceivably act on three different vectors.