100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

66 H. Nicolaiwhere the abelian part A belongs to the exponentiation of the CSA, andthe nilpotent part N to the exponentiation of n + . This formal Iwasawadecomposition, which is the infinite dimensional analog of (33), can bemade fully explicit by decomposing A and N in terms of bases of h and n +(using the Cartan Weyl basis)A(t) = exp ( β µ (t) H µ),( ∑N (t) = expmult(α)∑α∈∆ + s=1ν α,s (t) E α,s)(71)where ∆ + denotes the set of positive roots. The components β µ , parametrizinga generic element in the CSA h, will turn out to be in direct correspondencewith the metric scale factors β a in (34). The main technical differencewith the kind of Iwasawa decompositions used in section 3.1 is that nowthe matrix V(t) is infinite dimensional for indefinite g(A), in which case thedecomposition (71) is, in fact, the only sensible parametrization available!Consequently, there are now infinitely many ν’s, whence N contains aninfinite tower of new degrees of freedom. Next we definewithN˙N −1 = ∑mult(α)∑α∈∆ + s=1j α,s E α,s ∈ n + (72)j α,s = ˙ν α,s + “ν ˙ν + νν ˙ν + · · ·′′ (73)(we put quotation marks to avoid having to write out the indices). To definea Lagrangian we consider the quantity˙VV −1 = ˙β µ H µ + ∑mult(α)∑α∈∆ + s=1exp ( α(β) ) j α,s E α,s (74)which has values in the Lie algebra g(A). Here we have setα(β) ≡ α µ β µ (75)for the value of the root α ( ≡ linear form) on the CSA element β = β µ H µ .Next we defineP := 1 (˙VV −1 + (2˙VV )−1 ) T= ˙β µ H µ + 1 mult(α)∑ ∑j α,s exp ( α(β) ) (E α,s + E −α,s ) (76)2α∈∆ + s=1