Ricci fall-off in static and stationary non-singular spacetimes, revisited: the null geodesic method

Abstract

We revisit an older concept in singularity theory: that in the presence of the strong energy condition (SEC), a static or stationary spacetime must have a quadratic fall-off in a characteristic Ricci quantity, in order for the spacetime to be without singularities (or, at least, to be both globally hyperbolic and timelike or null geodesically complete). We replace SEC with the null energy condition (NEC), and apply the methods used previously on timelike geodesics, to null geodesics instead. The results are noticeably weaker for the NEC case than for SEC: using a somewhat different characteristic measure of Ricci curvature, we obtain a fall-off which is quadratic only if there is not much asymptotic change in the size of the Killing field: we employ a ratio of maximum size to square of minimum size of the Killing field—within a ball of given radius r—in addition to $1/r^{2}$.