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

The linear system 265where we consider the expansion around zero and infinity. What is thesignificance of the replacement (14.133)? A spectral parameter is required if onewants to enlarge the finite Lie group to its affine extension, and the appearance oft in (14.133) fits nicely with this expectation. There is now an infinite hierarchyof fields, as one can see by expanding ˆv in t. For convenience let us pick ageneralized triangular gauge, defined by the requirement that ˆv should be regularat t = 0, or∞∑ˆv(x; t) = exp t n ϕ n (x). (14.148)Another important feature of the linear system is the invariance under ageneralization of the symmetric space automorphism. Let us define it for η = 1n=0τ ∞ ˆv(t) = ( ˆv T ) −1 ( 1t). (14.149)In terms of the Lie algebra, the action of τ ∞ readsIt is straightforward to verify thatQ µ → Q µ , P µ →−P µ . (14.150)τ ∞ ( ˆv −1 ∂ µ ˆv) =ˆv −1 ∂ µ ˆv. (14.151)We can say that it is ˆv∂ µ ˆv ∈ H ∞ , which is the subalgebra of the Geroch groupG ∞ t which is t ∞ -invariant, as happens for finite-dimensional symmetric spaces.It is worth noticing that this property does not hold for v −1 ∂ µ v if we replace τ ∞with the transformation τ defined in section two.14.5.3 Derivation of the colliding plane metric by factorizationAt this point we can convince ourselves that the results obtained so far can beused to construct exact solutions of Einstein’s equations. Of central importancefor this task is the monodromy matrix, which is defined as follows(Å =ˆv(x; t) ˆv T x; 1 ). (14.152)tA short calculation reveals that∂ µ Å =ˆv(ˆv −1 ∂ µ ˆv − τ ∞ ( ˆv −1 ∂ µ ˆv))τ ∞ ˆv −1 = 0 (14.153)where the relation (14.151) was used. Consequently, Å can only depend on w.The solutions generating procedure now consists in choosing a matrix Å(w) andfinding a factorization as in (14.152).