Maximum Principles for Null Hypersurfaces and Null Splitting Theorems

The geometric maximum principle for smooth (spacelike) hypersurfaces, which is a consequence of Alexandrov’s [1] strong maximum for second order quasilinear elliptic operators, is a basic tool in Riemannian and Lorentzian geometry. In [2], extending earlier work of Eschenburg [7], a version of the geometric maximum principle in the Lorentzian setting was obtained for rough (C) spacelike hypersurfaces which obey mean curvature inequalities in the sense of support hypersurfaces. In the present paper we establish an analogous result for null hypersurfaces (Theorem 3.4) and consider some applications. For the applications, it is important to have a version of the maximum principle for null hypersurfaces which does not require smoothness. The reason for this, which is described in more detail in Section 3, is that the null hypersurfaces which arise most naturally in spacetime geometry and general relativity, such as black hole event horizons, are in general C but not C. To establish our basic approach, we first prove a maximum principle for smooth null hypersurfaces (Theorem 2.1), and then proceed to the C case. The general C version is then applied to study some rigidity properties of spacetimes which contain null lines (inextendible globally maximal null geodesics). The standard Lorentzian splitting theorem, which is the Lorentzian analogue of the Cheeger-Gromoll splitting theorem of Riemannian geometry, describes the rigidity of spacetimes which contain timelike lines (inextendible globally maximal timelike geodesics); see [3, Chapter 14] for a nice exposition. Here we show how the maximum principle for rough null hypersurfaces can be used to obtain a general “splitting theorem” for spacetimes with null lines (Theorem 4.1). We then consider an application of this null splitting theorem to asymptotically flat spacetimes. We prove that a nonsingular asymptotically flat (in the sense of Penrose [17]) vacuum (i.e., Ricci flat) spacetime which contains a null line is isometric to Minkowski space (Theorem 4.3). In Section 2 we review the relevant aspects of the geometry of null hypersurfaces and present the maximum principle for smooth null hypersurfaces. In