A no-go theorem for closed timelike curves in globally null-foliated spacetimes

Abstract

The study of closed timelike curves has proceeded one spacetime at a time, and the criteria available form three families, the time function criteria, the energy condition criteria, and the topological criteria. We initiate a fourth family of criteria, elementary in form and resting on a single scalar equation rather than any energy condition. We prove that a spacetime admitting a globally defined function whose gradient is everywhere null cannot contain a closed timelike curve, and that the hypothesis is logically independent of stable causality rather than weaker than it. The proof turns on one algebraic fact, that a null covector is never orthogonal to a timelike vector, so the function grows strictly along every timelike curve and cannot close. Because the null eikonal equation alone decides the criterion, we can use it where the classical time function theorem is silent. We establish for the first time that no Kundt spacetime and no Robinson--Trautman spacetime can contain a closed timelike curve on any region where the null coordinate is globally single valued, whatever the cosmological constant and the rotation one form, which settles a question left open for gyratonic waves. We further show that the Misner, Ori and Ahmed constructions reduce to the statement that a compact null direction has turned timelike, and that the cosmological constant, the expansion and the shear are spectators in all of them.