(ed.). Gravitational waves (IOP, 2001)(422s).

248 Infinite-dimensional symmetries in gravityAfter some algebra, we find abc = 1 2 ((3)abc − e [a m η b]c −1 ∂ m ) (14.17) 3 ab = 3/2 e m a e n b B mn (14.18) 3 3b =− 1 2 e b m −1/2 ∂ m (14.19) c 3b = 0. (14.20)Substituting the ansatz for the vierbein in the field action and making use of theabove decomposition of the Einstein action, after some calculations we arrive atthe following result− 1 4 ER(E) =−1 4 eR(3) (e) −16 1 e2 B mn B mn + 1 8 egmn −2 ∂ m ∂ n (14.21)where B mn = ∂ m B n − ∂ n B m .Duality transformationThe very special feature of three dimensions is that the Kaluza–Klein vector fieldcan be dualized to a scalar. This is achieved by adding to the Einstein–HilbertLagrangian the expressionÄ ′ = 1 8 e˜ɛmnp B mn ∂ p B (14.22)where B is a Lagrange multiplier and ˜ɛ mnp the Levi-Civita totally antisymmetricsymbol. The dualization makes the Lagrangian depend only on B. So, adding Ä ′to Ä and varying B n leads toe 2 B mn = ɛ mnp ∂ p B (14.23)modulo a numerical constant. Here we have set ɛ mnp = e˜ɛ mnp . Whenwe substitute this expression in the three-dimensional reduced Einstein–HilbertLagrangian, we get a new one with two scalar fieldsÄ =− 1 4 eR(3) (e) + 1 8 egmn −2 (∂ m ∂ n + ∂ m B∂ n B). (14.24)This is consistent with the equation of motion ∂ m (e 2 B mn ) = 0. In fact, the termwe add to the Lagrangian, which is now three dimensional, can be dropped by anintegration by parts and the use of the three-dimensional Bianchi identities for thetensor B mn .14.1.5 Dimensional reduction D = 3 → D = 2Then we perform a dimensional reduction from three to two, i.e. we have twoKilling commuting vectors (∂ 3 and ∂ 2 ) and there is no dependence on x 2 at all.x m = (x µ , x 2 ). (14.25)