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
between the electric field and the magnetic field .
In 4d, we have
The 4d constitutive relation is
which under assumptions of linearity gives
where . This may be written in a form that more closely mimics the tradition relations via
where (Note: 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.