Using ultra-high energy cosmic rays and air showers to test Lorentz invariance within modified Maxwell theory

Cosmic rays and air showers at ultra-high energy are unique tools to test the validity of Lorentz invariance. A brief overview is given on such tests focusing on isotropic, non-birefringent Lorentz violation (LV) in the photon sector. Based on the apparent absence of vacuum Cherenkov radiation and photon decay, the LV parameter $\kappa$ is bound to $-0.6 \cdot 10^{-20} < \kappa < 6 \cdot 10^{-20}$ (98\% CL). We report an updated limit from cosmic-ray photon observations and preliminary results on testing vacuum Cherenkov radiation in air showers.