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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CERN1
TL;DR: In this article, the supersymmetric extension of invariants quadratic in the Riemann curvature tensor for SO(N )-extended supergravity was studied.
43 citations
TL;DR: In this paper, it was shown that complete non-compact Riemannian manifolds with non-negative Ricci curvature of dimension greater than or equal to two in which some Gagliardo-Nirenberg type inequality holds are not very far from the Euclidean space.
43 citations
TL;DR: In this article, the authors prove that there is a set of initial data, open with respect to the C-2 X C-1 topology and dense with regard to C-infinity topology, such that the corresponding space-times have the following properties: given an inextendible causal geodesic, one direction is complete and the other is incomplete; the Kretschmann scalar, i.e., the Riemann tensor contracts with itself, blows up in the incomplete direction; and the Kretchmann tensor, the R
Abstract: This is the first of two papers that together prove strong cosmic censorship in T-3-Gowdy space-times In the end, we prove that there is a set of initial data, open with respect to the C-2 X C-1 topology and dense with respect to the C-infinity topology, such that the corresponding space-times have the following properties: Given an inextendible causal geodesic, one direction is complete and the other is incomplete; the Kretschmann scalar, ie, the Riemann tensor contracted with itself, blows up in the incomplete direction In fact, it is possible to give a very detailed description of the asymptotic behavior in the direction of the singularity for the generic solutions In this paper, we shall, however, focus on the concept of asymptotic velocity Under the symmetry assumptions made here, Einstein's equations reduce to a wave map equation with a constraint The target of the wave map is the hyperbolic plane There is a natural concept of kinetic and potential energy density; perhaps the most important result of this paper is that the limit of the potential energy as one lets time tend to the singularity for a fixed spatial point is 0 and that the limit exists for the kinetic energy We define the asymptotic velocity v(infinity) to be the nonnegative square root of the limit of the kinetic energy density The asymptotic velocity has some very important properties In particular, curvature blowup and the existence of smooth expansions of the solutions close to the singularity can be characterized by the behavior of v(infinity) It also has properties such that if 0 1 and v(infinity) is continuous at theta(0), then v(infinity) is smooth in a neighborhood of theta(0) Finally, we show that the map from initial data to the asymptotic velocity is continuous under certain circumstances and that what will in the end constitute the generic set of solutions is an open set with respect to the C-2 X C-1 topology on initial data
43 citations
Book•
15 Mar 2022TL;DR: In this article, the authors prove the global stability of the flat space among solutions of the Einstein-Klein-Gordon system in the case of small, smooth, and localized perturbations of the stationary Minkowski solution.
Abstract: We prove definitive results on the global stability of the flat space among solutions of the Einstein-Klein-Gordon system. Our main theorems in this monograph include: (1) A proof of global regularity (in wave coordinates) of solutions of the Einstein-Klein-Gordon coupled system, in the case of small, smooth, and localized perturbations of the stationary Minkowski solution; (2) Precise asymptotics of the metric components and the Klein-Gordon field as the time goes to infinity, including the construction of modified (nonlinear) scattering profiles and quantitative bounds for convergence; (3) Classical estimates on the solutions at null and timelike infinity, such as bounds on the metric components, weak peeling estimates of the Riemann curvature tensor, ADM and Bondi energy identities and estimates, and asymptotic description of null and timelike geodesics.
43 citations
TL;DR: In this paper, the authors analyzed the coupling of the electromagnetic field with the Riemann tensor and found that an amplification of the magnetic field occurs during the reheating phase of evolution of the universe.
Abstract: The inflationary amplification of magnetic field seeds for galaxies is discussed in the framework of the f(R) theories of gravity, where f(R) is of the form f(R)~Rn. The breaking of the conformal invariance necessary for the seeding of the primordial magnetic field from the vacuum is generated by means of the coupling of the electromagnetic field with curvature terms. We analyze the coupling of the electromagnetic field with the Riemann tensor, FαβFα'β'Rαβα'β'. We find that amplification of the magnetic field occurs during the reheating phase of evolution of the Universe. Estimates of the index n are derived by using the observed strength of the galactic magnetic fields. Moreover, the coupling of the electromagnetic field with a generic function of the scalar curvature, i.e. RmFαβFαβ, is discussed. In this case, we find that a growing of the primordial magnetic field during the reheating epoch may occur for an appropriate choice of the powers m and n.
43 citations