The Constant Mean Curvature Einstein flow and the Bel-Robinson energy
نویسنده
چکیده
We give an extensive treatment of the Constant Mean Curvature (CMC) Einstein flow from the point of view of the Bel-Robinson energies. The article, in particular, stresses on estimates showing how the Bel-Robinson energies and the volume of the evolving states control intrinsically the flow along evolution. The treatment is for flows over compact three-manifolds of arbitrary topological type, although the form of the estimates may vary depending on the Yamabe invariant of the manifold. We end up showing well posedness of the CMC Einstein flow with H ×H regularity, and proving a criteria for a flow to be a long-time flow on manifolds with non-positive Yamabe invariant in terms only of the first order Bel-Robinson energy.
منابع مشابه
Bel–robinson Energy and Constant Mean Curvature Foliations
An energy estimate is proved for the Bel–Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an L∞ bound on the second fundamental form, and a bound on a spacetime version of Bel–Robinson energy.
متن کاملSpacetimes admitting quasi-conformal curvature tensor
The object of the present paper is to study spacetimes admitting quasi-conformal curvature tensor. At first we prove that a quasi-conformally flat spacetime is Einstein and hence it is of constant curvature and the energy momentum tensor of such a spacetime satisfying Einstein's field equation with cosmological constant is covariant constant. Next, we prove that if the perfect flui...
متن کاملThe modification of the Einstein and Landau-Lifshitz pseudotensrs
Deser et al. proposed a combination of the Einstein and Landau-Lifshitz pseudotensors such that the second derivatives in vacuum are proportional to the Bel-Robinson tensor. Stimulated by their work, the present paper discuss the gravitational energy-momentum expression which has the same desirable Bel-Robinson tensor property. We find modifications of the Einstein and Landau-Lifshitz pseudoten...
متن کاملThe ground state and the long-time evolution in the CMC Einstein flow
Let (g,K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y (Σ)). As noted by Fischer and Moncrief, the reduced volume V(k) = (−k 3 )V olg(k)(Σ) is monotonically decreasing in the expanding direction and bounded below by Vinf = (−1 6 Y (Σ)) 3 2 . Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequen...
متن کاملSingular Isotropic Cosmologies and Bel - Robinson Energy 1
We consider the problem of the nature and possible types of spacetime singularities that can form during the evolution of FRW universes in general relativity. We show that by using, in addition to the Hubble expansion rate and the scale factor, the Bel-Robinson energy of these universes we can consistently distinguish between the possible different types of singularities and arrive at a complet...
متن کامل