ON THE RADIUS OF INJECTIVITY OF NULL HYPERSURFACES

摘要

This paper is concerned with the regularity properties of boundaries N(p) =(р) of the past (future) sets of points in a 3 + 1 Lorentzian manifold (M, g).The past of a point p, denoted Z— (p), is the collection of points that can be reached by a past directed time-like curve from p. As it is well known the boundaries N + (p)play a crucial role in understanding the causal structure of Lorentzian manifolds and the propagation properties of linear and nonlinear waves, e.g. in flat spacetime the null cone N— (p) U N+ (p) is exactly the propagation set of solutions to the standard wave equation with a Dirac measure source point at p. However these past (future) boundaries fail, in general, to be smooth even in a smooth, curved, Lorentzian space-time; one can only guarantee that N+(р) is a Lipschitz, achronal, 3-dimensional manifold without boundary ruled by in-extendible null geodesics from p; see [HE]. In fact N+(р) {p} is smooth in a small neighborhood of p but fails to be so in the large because of conjugate points, resulting in formation of caustics,or because of intersections of distinct null geodesics from p. Providing a lower bound for the radius of injectivity of the sets N + (p) is thus an essential step in understanding more refined properties of solutions to linear and nonlinear wave equations in a Lorentzian background.