Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1990. Sectio Physicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 20)

Anatol Nowicki: Composite spacetime from twistors and its extensions

- 12 Let the pair has the form C4a), therefore any equivalent pair of twistors satisfy [í,*] = CT..T.JM - [g g] iZ = QM I_ = ilM C6a3 where the 2x2 complex matrices fi,II are constructed of the coordinates of the twistors Therefore, we obtain iZ = fin"

1 <=> Ü = izn This is a Penrose relation in matrix form. Let us denote Cöb) CT T ) = o*

1 o

1

2 o*

1 o*

2 U11 n2t n22 •C) C7a) now, from (6b) we obtain /3i „a2 _ • „a/3_ O = (32 ct, ft =* 1,2 or more simply O iZ^T T = C7b> C7cD it is the incidence equation postulated first by Penrose. Its physical meaning: is the following CI 3: the point z <£ CM corresponds to the twistor T o =iz W/3 It is obvious that all twistors lying on the Z—plane given in the Cd) relations correspond to a given z g ÜT* point and for a given twistor T satisfying C7c) only one complex spacetime point z is assigned. If one needs to describe the real space—time point x € KM

4', one should require the matrix Z to be hermitean i.e. z = z

+ z = - ißrr

1 = icn1)

+ß

+ C8a>