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

The Twistor Approach to Space-Time Structures 489so if one vanishes so must the other, whence C abcd as a whole must vanish— the condition for (local) conformal flatness. It may be remarked, however,that in the positive-definite case (signature+ + ++) x and the splitsignaturecase (signature + + −−) y there is a large family of conformally(anti-)self-dual 4-manifolds. These spaces, and their twistor spaces, havea considerable pure-mathematical interest, z there being (different) “realityconditions” on each of the independent quantities Φ ABCD and X A′ B ′ C ′ D ′in these two cases.Yet, complex (anti-)self-spaces are by no means devoid of physical interest,especially those which can arise from deformations of (parts of) T forwhich F has the special form F = ε AB ∂f/∂ω A ∂/∂ω B , as given at the beginningof this section. We note that, in these cases, the operator ∂/∂π A ′ doesnot appear, and it follows that its infinitesimal action on Z α = (ω A , π A ′)leaves π A ′ unaffected. Thus, F generates a deformation of (part of) T thatpreserves the projectionF : T → ¯S ∗ ,(where ¯S ∗ is the complex conjugate of the dual S ∗ of the spin space S; notethat S is the space of 2-spinors like ω A , and ¯S ∗ is the space of those likeπ A ′). In each coordinate patch U i , with standard twistor coordinates, thisprojection takes the form(ω A , π A ′) ↦→ π A ′ ,and this now extends to a projection F that applies to the whole (nonprojective)curved twistor spaceF : T → ¯S ∗The inverse F −1 of this projection is a fibration of PT , each fibre being theentire complex 2-surface in T which projects down to a particular π in ¯S ∗ .In this case, the lines in PT can be neatly characterized as the projectiveversions of the holomorphic cross-sections of this fibration. These are theresults of maps R : ¯S∗ → T (whose composition F ◦ R with F is theidentity on ¯S ∗ ) which lift ¯S ∗ back into PT , and they generalize what in thecanonical flat case would be expressed as π A ′ ↦→ (ir AA′ π A ′, π A ′), where r ais the position vector of the point in CM that this cross-section defines.(Note the appearance of the basic incidence relation of §2.)The 2-surfaces of this fibration have tangent directions annihilated bya simple closed 2-form τ which is ( 1 2×) the exterior derivative of a 1-formι (of homogeneity degree 2):2τ = dι , ι ∧ τ = 0