K-surfaces in the Schwarzschild space-time and the construction of lattice cosmologies

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Abstract
We investigate spacelike spherically symmetric hypersurfaces of constant mean curvature K (which we call K‐surfaces) in spherically symmetric static spacetimes. We obtain the differential equation satisfied by these surfaces from a variational principle. The spacetime Killing vector leads to a first integral in the form of a conservation of energy for a particle moving in an effective potential. An embedding of the K‐surfaces’ intrinsic geometry in flat space likewise follows from an effective potential motion. We apply the formalism to the Schwarzschild solution, and display results of numerical integrations for a variety of K‐surfaces and their flat space embeddings. We use these to construct ’’lattice’’ cosmological models, and obtain a foliation of K‐surfaces of such models with large scale behavior of both the open and closed Friedmann type.