# The geometric origin of inhomogeneous media

## Summary

Just as geometry is encoded in the metric tensor and manifests itself via the Hodge star operator, so too do the electromagnetic constitutive equations?. For example, in linear media we have the simple constitutive relations

$E=\frac{1}{ϵ}D\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}B=\mu H.$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

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

The 4d constitutive relation is

$G=\star F\phantom{\rule{thinmathspace}{0ex}},$G = \star F \,,

which under assumptions of linearity gives

$\star F=-\eta \left(D-H\wedge dt\right)\phantom{\rule{thinmathspace}{0ex}},$\star F = -\eta(D-H\wedge d t) \,,

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

$\star \left(vdt\wedge E\right)=\frac{1}{ϵ}D\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\star B=\mu H\wedge vdt\phantom{\rule{thinmathspace}{0ex}},$\star(v d t\wedge E) = \frac{1}{\epsilon} D\quad\text{and}\quad\star B = \mu H\wedge v d t \,,

where $v=\frac{1}{\sqrt{\mu ϵ}}$ (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.

## References

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