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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: There is a unique Lagrangian quadratic in the curvature tensor which yields second-order field equations in dimensions greater than four as discussed by the authors, which is applied to a Kaluza-Klein theory.
Abstract: There is a unique Lagrangian quadratic in the curvature tensor which yields second-order field equations in dimensions greater than four. This Lagrangian is applied to a Kaluza-Klein theory and its cosmological implications are investigated.
44 citations
TL;DR: Katzin et al. as discussed by the authors introduced curvature collineations (CC), defined by a vector ξ, satisfying LξRbcda=0, where Rbcda is the Riemann curvature tensor of a Riemanian space Vn and L ξ denotes the Lie derivative, and they proved that a CC is related to a special conformal motion which implies the existence of a covariant constant vector field.
Abstract: Katzin et al. [G. H. Katzin, J. Levine, and W. R. Davis, J. Math. Phys. 10, 617 (1969)] introduced curvature collineations (CC), defined by a vector ξ, satisfying LξRbcda=0, where Rbcda is the Riemann curvature tensor of a Riemannian space Vn and Lξ denotes the Lie derivative. They proved that a CC is related to a special conformal motion which implies the existence of a covariant constant vector field. Unfortunately, recent study indicates that the existence of a covariant constant vector restricts Vn to a very rare special case with limited physical use. In particular, for a fluid space time with special conformal motion, either stiff or unphysical equations of state are singled out. Moreover, perfect fluid space times do not admit special conformal motions. This information was not available, in 1969, when CC symmetry was introduced. In this paper, CC is generalized to another symmetry called ‘‘curvature inheritance’’ (CI) satisfying LξRbcda=2αRbcda, where α is a scalar function. We prove that a proper...
44 citations
TL;DR: In this article, the authors derived a new exact static and spherically symmetric vacuum solution in the framework of the Poincare gauge field theory with dynamical massless torsion.
Abstract: We derive a new exact static and spherically symmetric vacuum solution in the framework of the Poincare gauge field theory with dynamical massless torsion. This theory is built in such a form that allows to recover General Relativity when the first Bianchi identity of the model is fulfilled by the total curvature. The solution shows a Reissner-Nordstrom type geometry with a Coulomb-like curvature provided by the torsion field. It is also shown the existence of a generalized Reissner-Nordstrom-de Sitter solution when additional electromagnetic fields and/or a cosmological constant are coupled to gravity.
44 citations
TL;DR: In this paper, the curvature tensor of the past (future) boundaries of points in regular, Einstein vacuum spacetimes has been investigated, and conditions, expressed in terms of a space-like foliation, have been derived for ensuring local nondegeneracy of these boundaries.
Abstract: We investigate the regularity of past (future) boundaries of points in regular, Einstein vacuum spacetimes. We provide conditions, expressed in terms of a space-like foliation and which imply, in particular, uniform bounds for the curvature tensor, sufficient to ensure the local nondegeneracy of these boundaries. More precisely we provide a uniform lower bound on the radius of injectivity of the null boundaries of the causal past (future) sets . Such lower bounds are essential in understanding the causal structure and the related propagation properties of solutions to the Einstein equations. They are particularly important in construction of an effective Kirchoff-Sobolev type parametrix for solutions of wave equations on . Such parametrices are used by the authors in a forthcoming paper to prove a large data break-down criterion for solutions of the Einstein vacuum equations
44 citations
TL;DR: In this article, it is shown that the left-hand side of the Regge equation may be interpreted geometrically as the sum of the moments of rotation associated with the faces of a polyhedral domain.
Abstract: In this paper the principle that the boundary of a boundary is identically zero (∂○∂≡0) is applied to a skeleton geometry. It is shown that the left-hand side of the Regge equation may be interpreted geometrically as the sum of the moments of rotation associated with the faces of a polyhedral domain. Here the polyhedron, warped though it may be, is located in a lattice dual to the original skeleton manifold. This sum is related to the amount of energy-momentum (E-p) associated to the edge in question. In the establishment of this equation the ordinary Bianchi identity is rederived by applying the principle that the (∂○∂≡0) in its (1–2–3)-dimensional formulation to polyhedral domain. Steps toward the derivation of the contracted Bianchi identity using this principle in its (2–3–4)-dimensional form are discussed. Preliminary results in this direction indicate that there should be one vector identity per vertex of the skeleton geometry.
“Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve.”—Ref. 3, Chap. 1.
44 citations