On a Gödel-like solution in non-relativistic gravity

Abstract 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 $$3+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> 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, $$D(x)=\cosh (mx)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo>cosh</mml:mo> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$H(x)=\sinh (mx)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo>sinh</mml:mo> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The resulting configuration describes a rotating non-relativistic universe supported by the constant potential term of the fluid Lagrangian and satisfies the identity $$D^{2}(x)-H^{2}(x)=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> throughout the entire spatial domain. As a consequence, the associated Killing vector remains spacelike everywhere and no closed timelike curves arise.