Spatiotemporal jumps as particular solutions in geodesic trajectories with the Gödel metric on an extended manifold

Abstract

Using the Gödel metric, we obtain some relevant solutions compatible with spatiotemporal jumps for the geodesic equations, by using an extension of General Relativity with nonzero boundary terms, which are described on an extended manifold generated by the connections $$\delta \Gamma ^{\mu }_{\alpha \beta } = b\,U^{\mu }\,g_{\alpha \beta }$$ δ Γ α β μ = b U μ g α β . These terms are given by a flow of velocities with components $$U^{\nu }$$ U ν : $$3\,b^2\,\nabla _{\nu }U^{\nu }=g^{\alpha \beta }\, \delta R_{\alpha \beta } = \lambda \left[ s\left( x^{\alpha }\right) \right] \,g^{\alpha \beta }\, \delta g_{\alpha \beta }$$ 3 b 2 ∇ ν U ν = g α β δ R α β = λ s x α g α β δ g α β in the varied Einstein–Hilbert action. The solutions are valid for an arbitrary equation of state with ordinary matter: $$\Omega =P/(c^2\,\rho ) = \frac{\left( \frac{\omega }{c}\right) ^2-\lambda (s)}{\left( \frac{\omega }{c}\right) ^{2}+\lambda (s)}$$ Ω = P / ( c 2 ρ ) = ω c 2 - λ ( s ) ω c 2 + λ ( s ) .