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

494 R. Penrosewhere we demand that the two given expressions for Σ are to be equal, thebilinear operator “∅” being defined byη∅(ρ ∧ σ) = (η ∧ ρ) ⊗ σ − (η ∧ σ) ⊗ ρas applied to any r-form η and 1-forms ρ and σ. We can express this ∅ in“index form” as twice the anti-symmetrization of the final index of η withthe two indices of the 2-form which follows the ∅ symbol. (This generalizesto a ∅-operation between an r-form and a t-form, where we take t× theantisymmetrization of the final index of the r-form with all indices of thet-form.) The preservation of the two quantities Π and Σ is really justasserting that on the overlap of two open sets U ′ and U, the quantities ιand θ must scale according to the (somewhat strange) rulesι ′ = κι , θ ′ = κ 2 θ , and dθ ′ = κ −1 dθ ,for some scalar function κ. The Euler homogeneity operator Υ (a vectorfield — see beginning of §3), which points along the Euler curves, can bedefined, formally, byΥ = θ ÷ φwhere we recall that the 4-form φ of §6 is defined from θ by 4φ = dθ. Moreprecisely, we can define Υ bydξ ∧ θ = Υ(ξ)φfor any scalar field ξ. We find, on the overlap between open regions U ′ andU (primed quantities κ ′ and Υ ′ referring to U ′ ), thatand, consequently,κ ′ = κ −1Υ ′ = κ 3 ΥΥ(κ) = 2κ −2 − 2κand, equivalently Υ(κ −1 ) = 2κ 2 − 2κ −1 . We can deduce from all this that,on overlaps, κ 3 takes the formκ 3 = 1 − f −6 (Z α )in standard flat-space terms, so we obtain the encoding of a twistor functionf −6 , homogeneous of degree −6 (in a fully cohomological way).This geometrical means of encoding a twistor function of the required“googly” homogeneity −6 may seem somewhat strange, where the informationis stored in a curious non-linear deformation of the scaling of the Euler