Space–Time Crystals: Reciprocity Classes, Forced Doublets and Defects

A sheet modulated in space and in time is a space–time crystal. Among its symmetries is the glide time reversal: a reflection of time combined with a shift by half a lattice vector. It acts on the response by transposition. This article states what it enforces in 2+1 dimensions. Every space–time group falls into one of four reciprocity classes. Plain reciprocity comes with a time mirror, glide reciprocity with a glide time reversal. A primed point operation gives a mapped relation instead, in which the sign of the magnetic dipole is measurable. On a sheet a threefold or sixfold axis rules out glide reciprocity and every Kramers-type doublet, and a fourfold axis allows the glide only in fixed half-vector classes. In a bulk crystal a threefold axis leaves the class along the axis. In the band structure the glide forces diabolic doublets, with loss and gain. At a momentum-gap edge it doubles the exceptional point. A horizontal cone of second-order media is protected by a symplectic square root of the period map. The energy–momentum zone is a Klein bottle only under a projective symmetry. In scattering the glide forbids the Littrow orders with phase -1 and the partner channel of a time slab. It gives twin pair-creation channels, a frequency-weighted Kirchhoff law and an effective medium without drag. Around a pump vortex of odd charge the parametric response leaves the half-integer sector. A sector rule fixes which defects lock, and above threshold a wall turns at a speed proportional to /N^2. Möbius closure forces a temporal defect, and edges must run along the glide. The results hold for linear sheets with a real, causal kernel. They are measured on lattice and sheet models, not on a fabricated metasurface.