Spacelike hypersurfaces with negative total energy in de Sitter spacetime

Abstract

De Sitter spacetime can be separated into two parts along two kinds of hypersurfaces and the half-de Sitter spacetimes are covered by the planar and hyperbolic coordinates, respectively. Two positive energy theorems were proved previously for certain \documentclass[12pt]{minimal}\begin{document}$\mathcal P$\end{document}P-asymptotically de Sitter and \documentclass[12pt]{minimal}\begin{document}$\mathcal H$\end{document}H-asymptotically de Sitter initial data sets by the second author and collaborators. These initial data sets are asymptotic to time slices of the two kinds of half-de Sitter spacetimes, respectively, and their mean curvatures are bounded from above by certain constants. While the mean curvatures violate these conditions, the spacelike hypersurfaces with negative total energy in the two kinds of half-de Sitter spacetimes are constructed in this short paper.