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On etudie l'evolution de l'energie de Bel-Robinson (BR) dans l'espace-temps T 3 ×R de Gowdy. On trouve qu'une quantite etroitement liee a l'energie de BR decroit de facon monotone quand l'espace-temps evolue vers la singularite finale, sauf dans le cas particulier des donnees de Kasner ou cette quantite est constante

We prove a global foliation result, using areal time, for T^2 symmetric spacetimes with a positive cosmological constant. We then find a class of solutions that exhibit AVTD behavior near the singularity.

We show here that if an analytic space‐time satisfying the vacuum (or electrovacuum) Einstein equations contains a compact null hypersurface with closed generators, then the space‐time must have a nontrivial Killing vector field. This result is an extension of an earlier theorem, which required that the generators of the hypersurface must not only be closed, but in addition must satisfy a local product bundle (LPB) condition. This LPB condition (which is known to be violated in certain of the Kerr–Taub–NUT space‐time models) is equivalent to the requirement that the generators must all be ordinary fibers of a Seifert fibration. We prove here that the LPB condition can be dropped. Thus we have stronger support for our conjecture that causality violations (as evidenced by Cauchy horizons) in cosmological solutions of the Einstein equations are essentially an artifact of symmetry, and are therefore nongeneric.

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.

In this paper we investigate whether classical (N=1) supergravity has a well‐posed locally causal Cauchy problem. We define well‐posedness to mean that any choice of initial data (from an appropriate function space) which satisfies the supergravity constraint equations and a set of gauge conditions can be continuously developed into a space‐time solution of the supergravity field equations around the initial surface. Local causality means that the domains of dependence of the evolution equations coincide with those determined by the light cones. We show that when the fields of classical supergravity are treated as formal objects, the field equations are (under certain gauge conditions) equivalent to a coupled system of quasilinear nondiagonal second‐order partial differential equations which is formally nonstrictly hyperbolic (in the sense of Leray–Ohya). Hence, if the fields were numerical valued, there would be an applicable existence theorem leading to well‐posedness. We shall observe that well‐posedness is assured if the fields are taken to be Grassmann (i.e., exterior algebra) valued, for then the second‐order system decouples into the vacuum Einstein equation and a sequence of numerical valued linear diagonal strictly hyperbolic partial differential equations which can be solved successively.