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.

Inconsistency of spin 4-spin 2 gauge field couplings

Abstract: The authors invert the usual problem of coupling higher spin gauge fields to gravity by treating (linearised) gravity as the 'matter field' source of spin 4 gauge theory. This is motivated by the existence of the conserved gravitational four-index Bel-Robinson tensor as a possible current for the spin 4 field. They first derive this tensor as a Noether current, thereby linking it to a novel invariance of spin 2. It is then shown that, as usual for higher spins, consistency does not survive beyond lowest-order cubic coupling.

Charged axially symmetric solution, energy and angular momentum in tetrad theory of gravitation

Abstract: Charged axially symmetric solution of the coupled gravitational and electromagnetic fields in the tetrad theory of gravitation is derived. The metric associated with this solution is an axially symmetric metric which is characterized by three parameters "the gravitational mass M, the charge parameter Q and the rotation parameter a." The parallel vector fields and the electromagnetic vector potential are axially symmetric. We calculate the total exterior energy. The energy–momentum complex given by Moller in the framework of the Weitzenbock geometry "characterized by vanishing the curvature tensor constructed from the connection of this geometry" has been used. This energy–momentum complex is considered as a better definition for calculation of energy and momentum than those of general relativity theory. The energy contained in a sphere is found to be consistent with pervious results which is shared by its interior and exterior. Switching off the charge parameter, one finds that no energy is shared by the exterior of the charged axially symmetric solution. The components of the momentum density are also calculated and used to evaluate the angular momentum distribution. We found no angular momentum contributes to the exterior of the charged axially symmetric solution if zero charge parameter is used.

Complexity Factor For Static Anisotropic Self-Gravitating Source in $f(R)$ Gravity

Abstract: In a recent paper, Herrera \cite{2} (L. Herrera: Phys. Rev. D97, 044010(2018)) have proposed a new definition of complexity for static self-gravitating fluid in General Relativity. In the present article, we implement this definition of complexity for static self-gravitating fluid to case of $f(R)$ gravity. Here, we found that in the frame of $f(R)$ gravity the definition of complexity proposed by Herrera, entirely based on the quantity known as complexity factor which appears in the orthogonal splitting of the curvature tensor. It has been observed that fluid spheres possessing homogenous energy density profile and isotropic pressure are capable to diminish their the complexity factor. We are interested to see the effects of $f(R)$ term on complexity factor of the self-gravitating object. The gravitating source with inhomogeneous energy density and anisotropic pressure have maximum value of complexity. Further, such fluids may have zero complexity factor if the effects of inhomogeneity in energy density and anisotropic pressure cancel the effects of each other in the presence of $f(R)$ dark source term. Also, we have found some interior exact solutions of modified $f(R)$ field equations satisfying complexity criterium and some applications of this newly concept to the study of structure of compact objects are discussed in detail. It is interesting to note that previous results about the complexity for static self-gravitating fluid in General Relativity can be recovered from our analysis if $f(R)=R$, which General Relativistic limit of $f(R)$ gravity. Some future research directions have been mentioned in the end of the summary.

On solutions of the Ricci curvature equation and the Einstein equation

Abstract: We consider the pseudo-Euclidean space (Rn, g), with n ≥ 3 and gij = δij ei, ei = ±1, where at least one ei = 1 and nondiagonal tensors of the form T = Σijfijdxidxj such that, for i ≠ j, fij (xi, xj) depends on xi and xj. We provide necessary and sufficient conditions for such a tensor to admit a metric ḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on Rn, on the n-dimensional torus Tn and on cylinders Tk×Rn-k, that solve the Ricci equation or the Einstein equation.