Lines in space-times (Q1200509)

let \((M,g)\) be a geodesically complete, time oriented Lorentz manifold and \(S\subset M\) be a compact, spacelike hypersurface without boundary. The authors show that there exists a complete, timelike geodesic that maximizes (Lorentzian) length between any two of its points, provided there exists a future complete, timelike geodesic starting at \(S\) that maximizes length between any of its points and \(S\).