Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 471where the zeroth order always corresponds to the undeformed quantity.Products of functions in  are simply mapped to ⋆−products of the correspondingfunctions in A. The same can be done for the action of thederivative ˆ∂ µ and consequently for an arbitrary differential operator actingon  [Aschieri et. al. (2005)].The fundamental dynamical field of our gravity theory is the vierbeinfield ʵ a . All other quantities such as metric, connection and curvaturecan be expressed in terms of it. Its image with respect to W −1 is denotedby E a µ . In the first approximation, we study the case E a µ = e a µ , wheree a µ is the usual vierbein field. Then for instance the metric is given up tosecond order in θ byG µν = 1 2 (E µ a ⋆ E ν b + E ν b ⋆ E µ a )η ab = 1 2 (e µ a ⋆ e ν b + e ν b ⋆ e µ a )η ab= g µν − 1 8 θα1β 1θ α2β 2(∂ α1 ∂ α2 e µ a )(∂ β1 ∂ β2 e ν b )η ab + . . . ,where g µν is the usual, undeformed metric. For the Christoffel symbol onefinds up to the second order [Meyer (2005)]:(i) the 0th order is the undeformed expressionΓ (0)ρµν = 1 2 [p µg νγ + p ν g µγ − p γ g µν ]g γρ ;(ii) the first order readsΓ (1)µν ρ = i 2 θαβ (∂ α Γ (0)σµν )g στ (∂ β g τρ ); and(iii) the second order isΓ (2)µν ρ = − 1 8 θα1β 1θ α2β 2((∂ α1 ∂ α2 Γ (0)µνσ)(∂ β1 ∂ β2 g σρ )−2(∂ α1 Γ (0)µνσ)∂ β1 ((∂ α2 g στ )(∂ β2 g τξ )g ξρ )−Γ (0)µνσ((∂ α1 ∂ α2 g στ )(∂ β1 ∂ β2 g τξ ) + g στ (∂ α1 ∂ α2 e τ a )(∂ β1 ∂ β2 e ξ b )η ab (3.292)−2∂ α1 ((∂ α2 g στ )(∂ β2 g τλ )g λκ )(∂ β1 g κξ ))g ξρ + 1 2 (∂ µ((∂ α1 ∂ α2 e aν )(∂ β1 ∂ β2 e bσ ))+ ∂ ν ((∂ α1 ∂ α2 e aσ )(∂ β1 ∂ β2 e bµ )) − ∂ σ ((∂ α1 ∂ α2 e aµ )(∂ β1 ∂ β2 e bν )))η ab g σρ ),where Γ (0)µνσ = Γ (0)ρµν g ρσ .