Maximal Hypersurfaces in Spacetimes with Translational Symmetry

Andrew Bulawa

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arXiv · 1603.00530

Maximal Hypersurfaces in Spacetimes with Translational Symmetry

Andrew Bulawa

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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.

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Andrew Bulawa. 2016-03-02. Maximal Hypersurfaces in Spacetimes with Translational Symmetry. https://arxiv.org/abs/1603.00530

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