nLab geometric origin of inhomogeneous media

Summary

Just as (Riemannian) geometry is encoded in the metric tensor and manifests itself via the Hodge star operator, so too do the constitutive equations of electromagnetism (cf. the discussion at pre-metric electromagnetism).

For example, in linear media we have the simple constitutive relations

$$E = \frac{1}{\epsilon} D \quad\text{and}\quad B = \mu H.$$

between the electric field $E$ and the magnetic field $B$.

In 4d, we have

$$F = B + E\wedge d t$$

The 4d constitutive relation is

$$G = \star F \,,$$

which under assumptions of linearity gives

$$\star F = -\eta(D-H\wedge d t) \,,$$

where $\eta = \sqrt{\frac{\mu}{\epsilon}}$. This may be written in a form that more closely mimics the tradition relations via

$$\star(v d t\wedge E) = \frac{1}{\epsilon} D\quad\text{and}\quad\star B = \mu H\wedge v d t \,,$$

where $v = \frac{1}{\sqrt{\mu\epsilon}}$ (Note: $v = c$ in vacuum).

What this means is the the electromagnetic properties of matter can be interpreted geometrically and are encoded in the Hodge star operator. Conversely, it means that geometrical properties of matter can be interpreted electromagnetically.

Related entries

• pre-metric electromagnetism

References

For more details see page 111 of Eric Forgy's dissertation.