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

The linear system 261Using the known commutation rules of the Chevalley–Serre generators ofSL(2, R) it is possible to directly check the algebra.For example, it is straightforward to see that[h 1 , e 0 ] = [T 3 0 , T −1 ] =−2T − 1 =−2e 0 (14.121)or that[e 1 , e 0 ] = [T0 + , T 1 − ] = 2T 13 (14.122)[e 1 , [e 1 , e 0 ]] =−4T1 + ⇒ [e 1[e 1 [e 1 , e 0 ]]] = 0and so on for the other commutators.The full current algebra is now built by taking multiple commutators in allpossible ways. The Lie algebra element c = h 0 + h 1 is the central charge. It hasa trivial action on the fields , B, B 2 , ˜λ, λ.14.5 The linear systemThe aim of this section is to linearize and localize the action of the Geroch groupseen in the previous section.Let us start from the Lagrangian for arbitrary G/H in three dimensions andthen reduce to twoÄ =− 1 4 ρeR(e) + 1 4 ρegmn Tr P m P n . (14.123)We pick now the conformal gauge for the three-bein, as before( )em a λδ = αµ 00 ρ(14.124)where we have dropped the two-dimensional Kaluza–Klein vector because itcarries no physical degrees of freedom any more. It is well known that the choiceeµ α = λeα µ is preserved under conformal diffeomorphismsδx + = ξ − (x + ), δx − = ξ + (x − ) (14.125)with the light cone coordinates x ± ≡ 1 √2(x 0 ± x 1 ). This residual coordinatefreedom can be gauged away, for example, by employing the dilaton and theaxion fields. One can fix the residual conformal diffeomorphisms by identifyingthe field ρ or ˜ρ with one of the coordinates.14.5.1 Solving Einstein’s equationsLet us now focus our attention on the way of solving Einstein’s equation. Firstnote that by substituting the gauge (14.124) into the scalar equation (14.67), wearrive atρ −1 D µ (ρ P µ ) = 0. (14.126)