Riemannian collineations in General Relativity and in Einstein-Cartan cosmology

Riemannian vectorial collineations along with current Killing conservation are shown to lead to tensorial collineations for the energystress tensor in general relativity and in Einstein-Cartan Weyssenhoff fluid cosmology. 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] Particle creation on chaotic inflationary models 2 In this note we show that the Riemannian collineations of the energy momentum tensor is obtained when the conserved Killing current is obtained from the beginning.Let us start by defining the vector Killing type current J = T abǫb (1) and By considering the Riemannian covariant derivative operator Da applied on equation (1) we obtain DaJ a = (DaT ab)ǫb + T abDaǫb (2) In the case ǫa is a Killing vector the following condition is obeyed D(aǫb) = 0 (3) Thus expression (3) yields automatically the conserved current since the energy-stress tensor in GR is conserved (DaT ) = 0.Therefore the conserved equation is immeadiatly given by DaJ a = 0 (4) Nevertheless in other theories like Einstein-Cartan the energy-stress is not automatically conserved like in GR,therefore one cannot use the Killing vector current on a trivial way.To include alternative gravity theories other than GR in our scheme we suggest that more general type of Riemannian or even non-Riemannian collineations can be used.Notice that if we assume current conservation from the beginning the expression (2) one obtains (DaT ab)ǫb = −T abDaǫb (5) Thus the recurrent spacetime [1] producing the Riemannian vector collineation Daǫb = C c ab ǫc (6) Substitution of (6) into (5) yields (DaT ac)ǫc = −C c ab ǫcT ab (7) As long as the recurrent vector ǫ is linearly independent equation (7) yields (DaT ac)ǫc = −C c ab T ab (8) Particle creation on chaotic inflationary models 3 Therefore the collineation coefficients C can be determined from this tensorial collineation.We shall now give a simple example in GR although formula (8) is equally valid in Einstein-Cartan or other type of alternative gravity.Let us consider the EMT of the type T ab = ρuu (9) of a dust fluid.Thus by applying (10) to (8) one is able to determine the coefficients C ab .Waldyr AQUI DeIXO essa demonstracao para o Shariff.Acho que em alguns dias faco uma demonstracao no caso de Einstein-Cartan, ok?Um abraco.Luiz Carlos Garcia de Andrade. ρr = 6πGσ 2 − [V + V ′ 3H + ηg2φ2 ] (10) By equating the RHS of equations (6) and (8) one obtains a differential equation for the potential V as 4V (φ) = [ V ′ 3H + ηg2φ2 ] (11) By solving equation (11) one obtains the following potential V (φ) = 36Hφ + 6Hηgφ + ηg 4 φ (12) which is clearly a potential for the chaotic inflation.Throughout the computations we assume the following approximation ηgφ << H.Finally by substituting expression (12) into equation (10) one obtains an expression for the spin-torsion density σ = ρr + 4φ+ 36H φ + ηg H (1 + g 36H + 6 H η )φ + ηg 4 φ (13) which shows that the spin-torsion density possess a kink potential term.A simple expression for the spin-torsion density perturbation in terms of the inflaton potential can be obtained by making use of the quantum fluctuation on the inflaton δφ = H 2 2π as δσ = δρr +2 H π + 36H π φ+ Hηg 4π (1 + g 36H + 6 H η )φ + Hηg 2π φ (14) A more detailed investigation on the matters disussed here may appear elsewhere. Particle creation on chaotic inflationary models 4