Spatiotemporal symmetries and energy-momentum conservation in uniform spacetime metamaterials

Spacetime metamaterials are opening new regimes of light-matter interactions based on the breaking of temporal and spatial symmetries, as well as intriguing concepts associated with synthetic motion. In this work, we investigate the continuous spatiotemporal translation symmetry of spacetime metamaterials with uniform modulation velocity. Using Noether's theorem, we demonstrate that such symmetry entails the conservation of the energy momentum. We highlight how energy-momentum conservation imposes constraints on the range of allowed light-matter interactions within spacetime metamaterials, as illustrated with examples of the collision of electromagnetic and modulation pulses. Furthermore, we discuss the similarities and differences between the conservation of energy-momentum and relativistic effects. We believe that our work provides a step forward in clarifying the fundamental theory underlying spacetime metamaterials.