On a G\"odel-like Solution in Non-Relativistic Gravity

Abstract

The article deals with G\"odel-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 $3+1$ spacetime. By adopting a G\"odel-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. Exact solutions are constructed by fixing the matter sector consistently with the field equations. The resulting configurations describe rotating non-relativistic universes and satisfy $D(x)>H(x)$ throughout the entire spatial domain. As a consequence, the associated Killing vector remains spacelike everywhere and no closed timelike curves arise.