Geodesics on non-Riemannian Geometric theory of Planar Defects

The method of Hamilton-Jacobi is used to obtain geodesics around nonRiemannian planar torsional defects.It is shown that by perturbation expansion in the Cartan torsion the geodesics obtained are parabolic curves along the plane x-z when the wall is located at the plane x-y.In the absence of defects the geodesics reduce to straight lines.The family of parabolas depend on the torsion parameter and describe a gravitationally repulsive domain wall.Torsion here plays the role of the Burgers vector in solid state physics. 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] The investigation of topological defects [1]in cosmology follows closed analogous models in solid state physics [2].Riemannian [3, 4, 5] and nonRiemannian [6, 7] models have been used to investigate these distributional defects such as disclinations [8] and dislocations [9] and domain walls analogous to ferromagnetic domains in crystallography.Lower dimensional models like disclinations in liquid crystals presented helical torsion structures as demonstrated by E.Dubois-Violette [10] and James Sethna [11].More recently F.Moraes [8] has been used the same method of Hamilton-Jacobi(H-J) equations to investigate geodesics around torsional defects like electrons around dislocated metals. In this Letter I follow this same patern to obtain geodesics around domain walls in 3-Dimensional gravity. Although isolated domain walls are nowadays of no interest in physics they still may play a role in mixed system in the condensed matter systems which serve as a laboratory for Cosmology.In this letter we find a simple example of a two dimensional gravity model from a domain wall solution of Einstein-Cartan gravity obtaing by reducing the dimensions of the spacetime metric ds = edt − e−Jz(dx2 + dy2)− dz (1) where J is the constant torsion at the planar wall,σ is the planar wall distribution of the wall energy density and G is the Newtonian gravitational constant. Due to translational symmetry of defects we shall consider the following metric ds = −e−Jzdx2 − dz (2) at the y = constant section. To obtain the geodesics I shall solve the H-J equation ∂W ∂t + H( ∂W ∂xi ) = 0 (3) where W [γ] = 1 2 ∫ t1 t0 gijẋ ẋdt (4) where γ is a parametrized curve where s[γ] = ∫ t1 t0 (gijẋ ẋ) 1 2 dt (5) Here H = L = 1 2 gijẋ ẋ is the Hamiltonian and L is the Lagrangian. We must stress that although in general Riemann-Cartan spaces test particles