Due to their extremely high particle energies (up to $10^{20}\,\text{eV}$), cosmic rays are ideally suited to search for violations of Lorentz invariance. We consider isotropic, nonbirefringent Lorentz violation in the photon sector and specialize to the case of a photon velocity larger than the maximum attainable velocity of standard Dirac fermions. Up to now, bounds on this type of Lorentz violation have been determined from observations of TeV gamma rays.

In this contribution, we present a novel approach to test Lorentz invariance with improved sensitivity. This approach is based on the measurement of extensive air showers which are induced by cosmic-ray particles in the Earth's atmosphere. We study the impact of Lorentz-violating processes (e.g. photon decay into an electron-positron pair) on the longitudinal development of air showers, in particular on the atmospheric depth of the shower maximum $X_\text{max}$. We use Monte Carlo simulations performed with the CONEX simulation code which was extended to include Lorentz-violating processes at a magnitude still allowed by present bounds. We show that taking into account these processes reduces the average $X_\text{max}$ for showers with primary energies above $10^{18}\,\text{eV}$ compared to standard physics by an amount that is much higher than the average resolution of current air shower experiments.