Periodic trajectories for a class of Lorentz-metrics of a time-dependent gravitational field (Q1823483)

We deal with the existence of timelike geodesics on the spacetime manifold \(({\mathbb{R}}^ 4,g)\), where the Lorentz tensor \(g=g(t,x)\) is T- periodic in t. By using a variant of a well-known curve shortening procedure, we show that there exists a timelike periodic trajectory on \(({\mathbb{R}}^ 4,g)\) i.e. a timelike geodesic \(z(s)=(t(s),x(s))\) such that \(x(s+1)=x(s)\) and \(t(s+1)=t(s)+T.\)