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.

The invariant classification of conformally flat pure radiation spacetimes

Abstract: We study conformally flat pure radiation spacetimes by means of their invariant classifications and show that no such spacetime requires higher than the fourth covariant derivative of the Riemann tensor in its invariant classification. Additional side results that we obtain are as follows. The Edgar - Ludwig metric for conformally flat pure radiation is shown to be a true generalization of the Wils metric; the subclass of the Edgar - Ludwig spacetimes which admit exactly one Killing vector is identified, generalizing Koutras's G1 subclass of the Wils metric; it is shown that the Edgar - Ludwig spacetimes with no Killing vectors can further be split into three discrete subsets, depending on the Cartan invariant where the fourth essential coordinate is found.

Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature

Abstract: The main result of this paper states that the traceless second fundamental tensor \({A}^{0}\)of an n-dimensional complete hypersurface M, with constant mean curvature H and finite total curvature, \(\int{M}^{|A^0|^n{d u}}{M} 0,\) any such surface must be compact.

The wave equation for the Lanczos potential. I

Abstract: The non-local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor C$\_{abcd}$ of Weyl. For this field Lanczos found a tensor potential L$\_{abc}$ with 16 independent components. We can make L$\_{abc}$ have only 10 effective degrees of freedom by imposing the 6 gauge conditions L$ \smallmatrix ab;s \\ s \endsmallmatrix $ = 0. Both fields C$\_{abcd}$, L$\_{abc}$ satisfy wave equations. The wave equation satisfied by L$\_{abc}$ is nonlinear, even in vacuo. However, a linear spinor wave equation for the Lanczos potential has been found by Illge but no correct tensor wave equation for L$\_{abc}$ has yet been published. Here, we derive a correct tensor wave equation for L$\_{abc}$ and when it is simplified with the aid of some four-dimensional identities it is equivalent to Illge's wave equation. We also show that the nonlinear spinor wave equation of Penrose for the Weyl field can be derived from Illge's spinor wave equation. A set of analogues of well-known results of classical electromagentic radiation theory can now be given. We indicate how a Green's function approach to gravitational radiation could be based on our tensor wave equation, when a global study of space-time is attempted.

Proof of the averaged null energy condition in a classical curved spacetime using a null-projected quantum inequality

Abstract: Quantum inequalities are constraints on how negative the weighted average of the renormalized stress-energy tensor of a quantum field can be. A null-projected quantum inequality can be used to prove the averaged null energy condition (ANEC), which would then rule out exotic phenomena such as wormholes and time machines. In this work we derive such an inequality for a massless minimally coupled scalar field, working to first order of the Riemann tensor and its derivatives. We then use this inequality to prove ANEC on achronal geodesics in a curved background that obeys the null convergence condition.

Second-Order Formalism for 3D Spin-3 Gravity

Abstract: A second-order formalism for the theory of 3D spin-3 gravity is considered. Such a formalism is obtained by solving the torsion-free condition for the spin connection ωaμ, and substituting the result into the action integral. In the first-order formalism of the spin-3 gravity defined in terms of SL(3, R) × SL(3, R) Chern–Simons (CS) theory, however, the generalized torsion-free condition cannot be easily solved for the spin connection, because the vielbein eaμ itself is not invertible. To circumvent this problem, extra vielbein-like fields ea(μν) are introduced as a functional of eaμ. New sets of affine-like connections ΓNμM are defined in terms of the metric-like fields, and a generalization of the Riemann curvature tensor is also presented. In terms of this generalized Riemann tensor the action integral in the second-order formalism is expressed. The transformation rules of the metric and the spin-3 gauge field under the generalized diffeomorphims are obtained explicitly. As in Einstein gravity, the new affine-like connections are related to the spin connection by a certain gauge transformation and a gravitational CS term expressed in terms of the new connections is also presented.