The solution space of a five-dimensional geometry: Kundt spacetimes and cosmological time-crystals

We uncover the solution space of a five dimensional geometry which we deem it as the direct counterpart of the Bianchi Type V cosmological model. We kinematically reduce the scale factor matrix and then, with an appropriate scaling and choice of time, we cast the spatial equations into a simple “Kasner” like form; thus revealing linear integrals of motion. Their number is enough so that, along with the quadratic constraint, it suffices to scan the entire space of solutions. The latter is revealed to be quite rich, containing cosmological solutions, some of which admit dimensional reduction asymptotically to four dimensions, Kundt spacetimes with vanishing type I (polynomial) curvature scalars and solutions describing periodic universes which behave like cosmological time crystals.