CMBR in Gödel Universe with torsion

A rotating universe represented by the Gödel metric in spacetimes with Cartan torsion is investigated where the Cosmic Microwave Background Radiation (CMBR) is computed from the Gödel rotation of the universe and the spin density of the spinning fluid in EinsteinCartan gravity.The autoparallel equation in Riemann-Cartan spacetime is shown to lead to the evolution equation of the cosmological perturbation where the spin-rotation coupling is the source of the growth of inhomogeneities.Unfortunatly the limit of the temperature anisotropy is far beyond the quantum capabilities of any experimental device and is therefore far away from any possibility of detection by PLANCK or MAP modern devices for measuring the CMBR. Departamento de F́ısica Teorica-UERJ. Rua São Fco. Xavier 524, Rio de Janeiro, RJ Maracanã, CEP:20550-003 , Brasil. E-Mail.: [email protected] Many years ago J.Silk [1] has discussed the stability of Gödel universe in the realm of general relativistic cosmology. Among other results he showed that the rotating universe is stable along the to perturbations in the plane of rotation while is unstable along the rotation axis;and that the CMBR temperature anisotropy depends on the Gödel rotation.In this letter we show that Silk result can be extended to Riemann-Cartan [2] spacetimes with torsion where use is made of the autoparallels [3]instead of geodesics in a manner that we are led to the evolution of the Gödel perturbation of densities in the spinning fluid in Einstein-Cartan gravity (EC).The idea of introducing the spin-torsion coupling with the Gödel rotation leads to a source coupling term between the spin and rotation.Solution of this equation allow us to investigate the stability of the Gödel universe with torsion.The evolution of the density perturbations allow us to write an expression for the anisotropy in the CMBR as done in GR the only and important diference is that now we are able to place an upper limit on torsion from the COBE satellite constraints.Let us now consider the Gödel metric in the form [4] ds = a[(dx) − (dx) + e 1 2 (dx) − (dx) + 2e 1 dxdx] (1) where a is the the Gödel scale parameter and Ω = (0, 0, c α √ 2 ).The autoparallel equation is given d ds v + [−Γμαβ + 2Q μ ( αβ)]v v = 0 (2) where Γμαβ is the Riemannian connection or the Christoffel connection and i = 1, 2, 3 and μ = 0, 1, 2, 3 and the torsion tensor Qμαβ = S μ αvβ is splitted in terms of the spin density tensor according to the Frenkel condition S αv α = 0 where the spin density tensor Sμα = −Sαμ.Taking the divergence of equation [2] yields d ds ∂μv μ + [−Γμαβ + 2Q μ ( αβ)]v v + 2[−∂Γμαβ + 2∂μQ μ ( αβ)](∂μv )v = 0 (3) From the conservation equation in Riemann-Cartan spacetime