On a Gödel-like solution in non-relativistic gravity

The article deals with Gödel-like solutions in the context of Galilean gravity, a geometric formulation of non-relativistic gravitation defined on a five-dimensional Galilean manifold. Within this framework, non-relativistic matter fields admit a covariant description, while the physical Newtonian dynamics is recovered through an immersion into the usual $$3+1$$ 3 + 1 spacetime. By adopting a Gödel-like metric ansatz and coupling the gravitational field to a Galilean fluid derived from a variational principle, we obtain a system of highly nonlinear and coupled field equations. This system is solved exactly: the matter sector is determined algebraically by the field equations, and the metric functions are obtained in closed form, D(x)= (mx) D ( x ) = cosh ( m x ) and H(x)= (mx) H ( x ) = sinh ( m x ) . The resulting configuration describes a rotating non-relativistic universe supported by the constant potential term of the fluid Lagrangian and satisfies the identity D²(x)-H²(x)=1 D 2 ( x ) - H 2 ( x ) = 1 throughout the entire spatial domain. As a consequence, the associated Killing vector remains spacelike everywhere and no closed timelike curves arise.