Some higher-dimensional vacuum solutions

We study an even-dimensional manifold with a pseudo-Riemannian metric with arbitrary signature and arbitrary dimensions. We consider the Ricci flat equations and present a procedure to construct solutions to some higher-(even-) dimensional Ricci flat field equations from the four-dimensional Ricci flat metrics. When the four-dimensional Ricci flat geometry corresponds to a colliding gravitational vacuum spacetime our approach provides an exact solution to the vacuum Einstein field equations for colliding gravitational plane waves in an (arbitrary) even-dimensional spacetime. We give explicitly higher-dimensional Szekeres metrics and study their singularity behaviour