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

Einstein theory 249Repeating the same steps as before, the two-bein now takes the form( ) (e a eµαρ Am =µ, e m eαµ−e0 ρ a =α σ A )σ0 ρ −1 . (14.26)Detailed calculation shows thatwith− 1 4 e(3) R (3) =− 1 4 ρeR(2) − 116 ρ3 eA µν A µν (14.27)eR (2) =−2∂ µ (ee αµ αβ β ) (14.28)where e is the determinant of the two-bein. At this point we can write theequations of motion for the theory. The equation of motion for the Kaluza–Kleinvector is given by∂ µ (ρ 3 eA µν ) = 0. (14.29)In two dimensions, a Maxwell field does not propagate, as there are notransverse degrees of freedom. Neglecting topological effects (i.e. non-vanishingholonomies) we can, therefore, set A µ = 0.For the remaining equations of motion, we can fix the gauge, and thencalculate them in a particular gauge, called the conformal gauge. The termeR (2) (e) is Weyl-invariant. To see why this is so, let us consider the termAn integration by parts gives− 1 4 ρ R(2) = 1 2 ρ∂ ν(ee α ν αγ γ ). (14.30)− 1 4 ρ R(2) ˙=− 1 2 eαγ γ e α µ ∂ µ ρ. (14.31)Then, using the definition of the anholonomy, we get= − 1 2 e(e α ν e τ γ ∂ ν e γ τ − e ν γ e τ α ∂ ν e γ τ )e αµ ∂ µ ρ (14.32)= − 1 2 egµν e τ γ ∂ ν e γ τ ∂ µ ρ − 1 2 e∂ νe ν α e αµ ∂ µ ρ (14.33)where another integration by parts and the definition of the two-bein have beenused.Now we can set the gauge, i.e. the 2D diffeomorphisms, by a condition onthe two-bein. So we writee α αµ = λẽ µ (14.34)with det ẽ µ α = 1 and λ = λ(x); hence, we are not considering the whole groupGL(2, R) but only its restriction to unimodular matrices SL(2, R).As we said before, we can set a particular gauge, the conformal gauge, byimposing the following condition on the two-bein. It is given byẽ µ α = δ α µ . (14.35)