The failure of the Ehlers--Kundt conjecture in the impulsive case

In 1962, Ehlers and Kundt conjectured that plane waves are the only class of complete Ricci-flat~\emph{pp}-waves, i.e.\ metrics on ${\mathbb{R}}^4$ of the form $$d{s}^2=2dudv+d{x}^2+d{y}^2+H(x,y,u)d{u}^2\mathrm{.}$$ Recently, Flores and Sánchez gave a proof of the conjecture in the fundamental case of spatially polynomially bounded profile functions $H$. However, impulsive pp-waves, i.e., waves with concentrated profile functions of the form $H(x,y,u)=f(x,y)\delta (u)$ ($\delta$, the Dirac measure) have been found to be complete. We summarise completeness results for several classes of impulsive wave spacetimes achieved during the last years and discuss them in the context of the Ehlers–Kundt conjecture.