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

Abstract In 1962, Ehlers and Kundt conjectured that plane waves are the only class of complete Ricci-flat pp -waves, i.e. metrics on $${\mathbb {R}}^4$$ R 4 of the form $$ ds^2=2du\,dv+dx^2+dy^2+H(x,y,u)du^2\,. $$ d s 2 = 2 d u d v + d x 2 + d y 2 + H ( x , y , u ) d u 2 . 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 $$H(x,y,u)=f(x,y)\,\delta (u)$$ H ( x , y , u ) = f ( x , y ) δ ( u ) ( $$\delta $$ δ , the Dirac measure) have been found to be complete for arbitrary (smooth) spatial profile functions f . 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.