On the Riemann Curvature Tensor in General Relativity

Abstract

The general theory of relativity is a theory of gravitation in which gravitation emerges as the property of the space-time structure through the metric tensor g ij . The metric tensor determines another object (of tensorial nature) known as Riemann curvature tensor. At any given event this tensorial object provides all information about the gravitational field in the neighbourhood of the event. It may, in real sense, be interpreted as describing the curvature of the space-time. The Riemann curvature tensor is the simplest non-trivial object one can build at a point; its vanishing is the criterion for the absence of genuine gravitational fields and its structure determines the relative motion of the neighbouring test particles via the equation of geodesic deviation.The above discussion clearly illustrates the importance of the Riemann curvature tensor in general relativity and it is for these reasons that a study of this curvature tensor has been made here. §2. The decomposition of Riemann tensor Using the decomposition of Riemann tensor and the invariants of Riemann tensor, a criterion for the existence of gravitational radiation has been given.The Riemann curvature tensor R k ijl is defined, for a covariant vector field A k , through the Ricci identitywhereThe Riemann curvature tensor can be decomposed aswhere C ijkl is the Weyl tensor, E ijkl = − 1 2 (g ik S jl + g jl S ik − g il S jk − g jk S il ), is the Einstein curvature tensor and S ij ≡ R ij − 1 4 g ij R, G ijkl ≡ − R 12 (g ik g jl − g il g

On harmonicity and Miao–Tam critical metrics in a perfect fluid spacetime

“…In Ahsan and Siddiqui [6] showed that a perfect fluid spacetime has divergencefree projective curvature tensor if and only if the Ricci tensor S is of Codazzi type. Also, Ahsan and Ali [3] proved that for a perfect fluid spacetime M with divergencefree projective curvature tensor, if the Ricci tensor S is covariantly constant, then M is of constant curvature and, respectively, if the energy-momentum tensor T is of Codazzi type, then r À 3p is constant.…”

Análisis de la estructura de singularidades en relatividad general mediante cálculo simbólico de los invariantes de curvatura

“…El tensor de curvatura (Tensor de Riemann), R , es un tensor del tipo (1,3), cuyas componentes son R a bcd , se puede descomponer de forma única en partes que son representaciones irreducibles del grupo completo de Lorentz…”

“…# esto ayuda a no visualizar la dependencia en las variables utilizadas Variedad > g := evalDG(−A(t) dX [1] &t dX [1] + B(t) dX [2] &t dX [2] + r 2 (dX [3] &t dX [3] + sin 2 (θ) dX [4] &t dX [4])) # Con este comando dotamos una métrica a nuestra variedad creada y &t denota el producto tensorial. Variedad > A sol := (simpli f y@value@eval@subs)(B sol , sol [2][]) # sustituimos la solución B dentro de la solución A con (eval@subs);…”

“…Variedad > Rie up := RaiseLowerIndices(gsolinv, Rie down , [1,2,3,4]) : # de la expresión resultante subimos todos sus índices Variedad > ContractIndices(Rie down &tRie up , [ [1,5], [2,6], [3,7], [4,8]])…”

“…# Definamos las funciones declaradas ContractIndices(C ab &tC ab , [ [1,7], [2,8], [3,5], [4,6]]) # Se ejecuta la acción deseada end proc:…”