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

TL;DR

This paper demonstrates that the metric of a massless particle exhibits an $E(2) imes r$ symmetry group with four Killing vectors, aligning with flat-space kinematics and extending beyond previous classifications of gravitational pp-waves.

Contribution

It reveals the symmetry structure of the massless particle metric obtained from boosting Schwarzschild, showing it has four Killing vectors and extends the JEK classification of pp-waves.

Findings

The massless particle metric has four Killing vectors.

The symmetry group is $E(2) imes r$, consistent with flat-space expectations.

The results extend the classification of gravitational pp-waves.

Abstract

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 $E(2)\times \RR$ 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}.