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Extended solution space for Chern-Simons gravity

Extended solution space for Chern-Simons gravity

EC + CS•

EC + CS• Einstein-Cartan rationaleδe c : ǫ abcd R ab (ω) ∧ e c = 0,δω a b : ǫ abqp e q ∧ T p + 2βdϑ ∧ R ab (ω) = 0.• The Bianchi identities together with the eoms → consistency conditions that generalize thePontryagin constraint.• ϑ acts as a source of torsion and torsion is highly constrained by experiments Kostelecký,Russell, Tasson PRL (2008). It is reasonable to expect the CS term to act as a smallperturbation to GR.• We propose the following perturbative Ansätze:e µ c = e (0)µ c + β e (1)µ c + ...ω µa b = ω (0) aµ b + β ω(1) aµ b + ... ,VIII Silafae - UTFSM, Valparaíso, 2010 Mauro Cambiaso New solutions in Chern-Simons gravity – p. 11

EC + CS: Solutions• Solving the field equations to first order in β we arrive at:ϑ();λ;αe (0) ǫ λδγ µR (0) αδγν+ ǫ λδγ νR (0) αδγµ(1) ;α (1) ;α (1) ;α= g να;µ + g µα;ν − g µν ;α − g (1) ;µ;ν .Curvature Riemann and metric are built with the tetrad to the corresponding order in β.• Semi-colon: zeroth-order covariant derivative, i.e. with the torsion-free connection.• For v µ = (1/m,0,0,0) and g (0)µν := Schwarzschild, LHS vanishes. Then g (1)µν = 0satisfies the RHS and thus Schwarzschild is a O(β) solution. ⇒ Classical tests OK!• Consistency conditions are all satisfied to O(β).• The latter provides a means to find new solutions as perturbations to a previously existingone, in a systematic way to all orders in β.VIII Silafae - UTFSM, Valparaíso, 2010 Mauro Cambiaso New solutions in Chern-Simons gravity – p. 12

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