Reciprocity and the adjoint medium – what the space-time symmetry preserves when the constitutive relation is not a scalar

A medium whose properties are modulated in space and time can be built to pass a wave one way and block it the other, and a body of work exists on how to do so. The opposite question is asked less often: what forbids it. For a medium described by a real scalar permittivity, one answer is known: if the modulation is unchanged by a shift of half a cell in space combined with a reversal of time, the band structure is symmetric in the direction of propagation and no isolation is possible. Real media are not scalars. They are described by tensors, they respond to the magnetic field as well as the electric one, and their response depends on frequency. We show that for such a medium the symmetry does not preserve reciprocity at all. What it does is tie the spectrum in the reversed direction to the spectrum of the adjoint medium — the medium obtained by transposing the constitutive tensors in the classical sense of the Lorentz reciprocity theorem. The medium is therefore reciprocal exactly when it is self-adjoint, which is the classical condition on the magnetoelectric coupling. Two conditions are needed and neither suffices alone: the symmetry does not make a medium reciprocal, and a self-adjoint medium without the symmetry is not reciprocal either. Both halves of that statement are measured. For a medium that is not self-adjoint the conclusion is not simply lost. In the nonreciprocal bi-isotropic class the spectrum in the reversed direction is exactly the spectrum of the medium with its magnetoelectric parameter reversed, so the asymmetry is described completely by a single sign. Self-adjointness is a property a medium either has or has not, so the criterion by itself says nothing about a medium that nearly has it; the departure from it turns out to be priced linearly, over a range located from the data rather than assumed. The whole apparatus survives dispersion. Carrying the polarisation as a second field keeps the equations local in time, and the argument transfers to the coupled system with one route lost and accounted for. An extension to a two-dimensional lattice is measured for the step that discriminates and stated as unmeasured for the rest. Numerical verification accompanies every claim; each measuring test is paired with an input on which the claim must fail, and the scripts are deterministic and deposited with this record. The constitutive relation treated here is local, and magnetoelectric coupling that arises as weak spatial dispersion lies outside it; the boundary is stated rather than left to be discovered.