Godel-type universes in bumblebee gravity

The bumblebee field coupled with gravity is considered. This gravitational theory exhibits spontaneous breaking of Lorentz symmetry. The Godel-type universe is introduced and then the causality and its violation are studied. Causal and noncausal Godel-type solutions are obtained for different content of matter. In addition, when the coupling constant, which controls the interaction between the bumblebee field and the gravitational field, is zero the bumblebee potential may be associated with the value of the cosmological constant of the general relativity. Furthermore, the case with the nonzero coupling constant is also investigated.