Rigid Singularity Theorem in Globally Hyperbolic Spacetimes

30 years ago, Penrose-Hawking have shown that spacetimes are geodesically incomplete under some physically reasonable conditions [1] [2] [3] [4]. The generic condition is the key assumption to induce singularities rigidly. Geroch improved these theorems with “no observer horizon” condition in place of the generic condition for the spatially closed universe [5,6]. Here, the “no observer horizon” condition is that for some p ∈ M , the set M \ [I(p) ∩ I−(p)] is compact (This definition is what Bartnik improved the original Geroch’s definition [7]). Under this assumption Geroch proved that a globally hyperbolic spacetime which has compact acausal hypersurfaces without edge (i.e. compact Cauchy surfaces) is geodesically incomplete or Ricci flat. In this statement flatness implies that the spacetime is static, i.e. a timelike Killing vector field is orthogonal to a family of spacelike hypersurface, (of course, there exists a static and non-flat spacetime, e.g. the Einstein static universe). Unfortunately, The Geroch’s proof is incomplete since it relies on erroneous assertions of Avez. The correction for the error was established by Bartnik [7]. Galloway modified Geroch’s theorem [8]. He investigated various equivalent forms of a “no observer horizon” type condition. Bartnik conjectured that the spacetime splits isometrically as space × time, i.e. static if spacetimes, which are timelike geodesically complete and globally hyperbolic with compact Cauchy surfaces, satisfy the timelike convergence condition (the strong energy condition) [7]. Eschenburg and Galloway proved the Bartnik’s conjecture with the modified “no observer horizon” condition, which is I−(γ) ⊃ S with S and γ being a compact partial Cauchy surface and an S-ray [9], instead of global hyperbolicity. Furthermore, Galloway and Horta proved it with the condition which is I−(γ) ∩ I(β) 6= ∅ with γ and β being future and past S-rays [10]. This result is closely related to Yau’s Lorentzian splitting conjecture [11]. This conjecture is proved by Eschenburg [12], Galloway [13], Newman [14] and recently Galloway and Horta [10], which reads,