A Note on the Symmetries of the Gravitational Field of a Massless Particle

It is shown that the metric of a massless particle obtained from boosting the Schwarzschild metric to the velocity of light, has four Killing vectors corresponding to an $E(2)\times \text{RR}$ symmetry-group. This is in agreement with the expectations based on flat-space kinematics but is in contrast to previous statements in the literature \cite{Schueck}. Moreover, it also goes beyond the general Jordan-Ehlers-Kundt-(JEK)-classification of gravitational pp-waves as given in \cite{JEK}.