How does $\epsilon \mu = 1/c_m^2$ change when $\mu$ or $\epsilon$ (permeability or permittivity) is a tensor?

I've read that in some materials, $\mu$ can be a tensor, not a mere scalar. (I haven't actually dealt with tensors before, but I'm assuming for my purposes here, it is synonymous with "matrix".)

I'm not sure if the same holds for $\epsilon$, but I'm assuming it might there too.

I'm fine with either of those, but then I'm wondering what the formula

$$\epsilon\,\mu = \frac{1}{c_m^2}$$

turns into when dealing with tensors? ($c_m$ being the speed of light in the medium)

1 Answer

If $\varepsilon$ or $\mu$ are tensors (read, matrices), then so is $c_m$: $$\overbrace{\varepsilon}^\mathrm{matrix} \underbrace{\mu}_\mathrm{matrix}=c_m^{-2}\ \leftarrow\ \text{matrix as well}$$

In other words, if the permeability and/or permittivity are matrices, then the speed of light is a matrix as well. In this case, the $\_^{\color{red}{-1}}$ is understood in the sense of matrix inverse.

In the coordiante system where $c_m$ is diagonal, we have $$c_m=\begin{pmatrix} c_1&\cdot&\cdot\\\cdot&c_2&\cdot\\\cdot&\cdot&c_3\end{pmatrix}$$ where $c_i$ is the speed of light along the $x_i$ axis.

So yes: the speed of light in a medium can be direction-dependent; see Birefringence.

• +1 Wow, I didn't expect this one, but it makes sense! I interpret this to mean that the speed of light is different in each direction? – Mehrdad Dec 26 '16 at 19:48
• @Mehrdad yes, that's it! see the link in the post for more details. – AccidentalFourierTransform Dec 26 '16 at 19:52
• Still, that is only the effective speed of light, accounting for different material properties in different directions. In between collisions and interactions with the material molecules etc the speed is always the same, c. – Bob Bee Dec 28 '16 at 2:14