Double cosmic walls in Teleparallel Gravity

An example is given of a plane topological torsion defect representing a cosmic wall double wall in teleparallel gravity.The parallel planar walls undergone a repulsive gravitational force due to Cartan torsion.This is the first example of a non-Riemannian double cosmic wall.It is shown that the walls oscillate with a speed that depends on torsion and on the surface density of the wall.Cartan torsion acts also as a damping force reducing the speed of oscillation when it is stronger. PACS numbers : 0420,0450. Departamento de F́ısica Teórica Instituto de F́ısica UERJ Rua São Fco. Xavier 524, Rio de Janeiro, RJ Maracanã, CEP:20550-003 , Brasil. e-mail.:[email protected] A solution of Einstein equations representing parallel planar topological defects [1] have been investigated in detail by Letelier [2].Although Riemannian topological defects are defined as the spacetime manifolds where the Riemann-Christoffel curvature tensor vanishes only off the defects, we here define teleparallel defects as solutions of the teleparallel condition R jkl(U4) = 0 (1) and have the same form of the metric of the Riemannian defect.In this Letter we shall consider the planar parallel cosmic walls in the context of the teleparallel theory of gravity as was developed by Einstein himself during the period in 1928 in Berlin [3].Let us consider the double cosmic wall solution as [2] ds = (ω) − (ω) − (ω) − (ω) (2) where the basis 1-forms ω (i=0,1,2,3) are given by ω = e 2 −z2)dt ω = 2ze 2 −z2)dz ω = e 2 −z2−t)dx ω = e 2 −z2−t)dy (3) where σ is the surface matter density of the wall and h is the separation between the walls.Here since we shall consider that the walls may move relative to each other this quantity will depend of the coordinates t and z.In the case of Letelier double Riemannian cosmic wall this amount is fixed.The torsion 1-forms are chosen such that the connection 1-forms ω j vanish.Thus from the second Cartan equation of differencial exterior forms Rk = dω i k + ω i l ∧ ω l k (4) where the curvature 2-form is R j = R i jklω k ∧ ω, we see that the teleparallel condition (1) is fulfilled.From the first Cartan equation T i = dω + ωk ∧ ω k (5)