  In extensive air showers induced by ultra-high energy (UHE) cosmic rays,
secondary particles are produced with energies far above those accessible by
other means. These extreme energies can be used to search for new physics. We
study the effects of isotropic, nonbirefringent Lorentz violation in the photon
sector. In case of a photon velocity smaller than the maximum attainable
velocity of standard Dirac fermions, vacuum Cherenkov radiation becomes
possible. Implementing this Lorentz-violating effect in air shower simulations,
a significant reduction of the calculated average atmospheric depth of the
shower maximum $\left<X_\text{max}\right>$ is obtained. Based on
$\left<X_\text{max}\right>$ and its shower-to-shower fluctuations
$\sigma(X_\text{max})$, a new bound on Lorentz violation is derived which
improves the previous one by a factor of 2. This is the first such bound based
on the absence of vacuum Cherenkov radiation from fundamental particles
(electrons and positrons) in air showers. Options for further improvements are
discussed.
