Maximal Hypersurfaces in Spacetimes with Translational Symmetry

TL;DR

This paper investigates maximal hypersurfaces in four-dimensional vacuum spacetimes with translational symmetry, establishing geometric classifications, positivity of mass, and properties of lapse functions in quotient spacetimes.

Contribution

It provides new results on the geometry and mass positivity of maximal hypersurfaces in spacetimes with translational symmetry, including classifications and lapse function behavior.

Findings

Complete noncompact maximal hypersurfaces are either flat cylinders or conformal to the Euclidean plane.

Positivity of mass is established for certain maximal hypersurfaces in the quotient setting.

Lapse functions are necessarily unbounded in nontrivial quotient spacetimes with maximal hypersurfaces.

Abstract

We consider four-dimensional vacuum spacetimes which admit a nonvanishing spacelike Killing field. The quotient with respect to the Killing action is a three-dimensional quotient spacetime $(M,g)$. We establish several results regarding maximal hypersurfaces (spacelike hypersurfaces of zero mean curvature) in such quotient spacetimes. First, we show that a complete noncompact maximal hypersurface must either be a cylinder $S^1 \times \mathbb{R}$ with flat metric or else conformal to the Euclidean plane $\mathbb{R}^2$. Second, we establish positivity of mass for certain maximal hypersurfaces, referring to a analogue of ADM mass adapted for the quotient setting. Finally, while lapse functions corresponding to the maximal hypersurface gauge are necessarily bounded in the four-dimensional asymptotically Euclidean setting, we show that nontrivial quotient spacetimes admit the maximal hypersurface gauge only with unbounded lapse.