A simple Godel-type universe

Einstein's field equations not only allow realistic solutions, but also allow exotic ones. In the first class, there are solutions describing FLRW universe, black holes and the gravitational waves. In the second class, we point out de Sitter and Anti-de Sitter universes without matter, wormholes solutions and Godel universe without time. In this paper, we are interested in finding out a simpler example of Godel-type universe. The road to the desired result passes through a spacetime without matter and time at the same time, whose texture is "flicked" by planar gravitational waves. This example allows us to create a simpler Godel-type universe.