Topic
Riemann curvature tensor
About: Riemann curvature tensor is a research topic. Over the lifetime, 6248 publications have been published within this topic receiving 138871 citations. The topic is also known as: Riemann–Christoffel tensor.
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TL;DR: In this paper, it has been shown that van Stockum's spacetime is free from singularities and matter at radial infinity below a certain value of the mass per unit length.
Abstract: The solution of van Stockum (1937) consists of a rotating dust interior, and three exterior metrics referring to different ranges of the mass per unit length. It has been stated in the literature that the exterior in static, but it is proved here that this is so only in the low-mass case. An examination of the Riemann tensor shows that van Stockum's spacetime is free from singularities and matter at radial infinity below a certain value of the mass per unit length. The ultrarelativistic case which has closed timelike lines, can occur at physically possible densities and radii.
107 citations
TL;DR: In this article, the super-Hamiltonian of an arbitrary tensor field was shown to split into two parts, H φ ↑ and Hπ −, with the former being local in the field momenta and the latter containing their first derivatives.
Abstract: Various kinematical relations, holding between hypersurface projections of spacetimetensor fields in an arbitrary Riemannian spacetime, are studied in terms of differential geometry in hyperspace. A criterion is given that a collection of hypertensor fields is generated by the projections of a single spacetimetensor field intersected by the embeddings. From here, it is shown that the super‐Hamiltonian of an arbitrary tensor field splits into two parts, H φ ↑ and H φ −, H φ ↑ being local in the field momenta and H φ − containing their first derivatives. The form of H φ − for an arbitrary tensor field is determined from the field behavior under hypersurface tilts. The kinematical equations for the intrinsic metric and the extrinsic curvature are written in a quasicanocical form, and their connection with the closing relations for the gravitational super‐Hamiltonian is exhibited. The conservation laws of charge, energy and momentum, and the contracted Bianchi identities, are written as hypertensor equations.
107 citations
TL;DR: In this article, a conformal deformation involving a fully nonlinear equation in dimension 4 was presented, starting with a metric of positive scalar curvature, and a conformally invariant condition for positivity of the Paneitz operator.
Abstract: We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with a metric of positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. A special case of this deformation provides an alternative proof to the main result in Chang, Gursky & Yang, 2002. We also give a new conformally invariant condition for positivity of the Paneitz operator, generalizing the results in Gursky, 1999. From the existence results in Chang & Yang, 1995, this allows us to give many new examples of manifolds admitting metrics with constant Q-curvature.
107 citations
TL;DR: In this paper, Sturm et al. introduced and studied rough curvature bounds for discrete spaces and graphs, and showed that the metric measure space which is approximated by a sequence of discrete spaces with rough curvatures ⩾ K will have curvature K in the sense of [J. Lott, C.Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. I, Acta Math.
107 citations