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

- 11 z 4 * z = z^o^ C2) One can get to the real Minkowski space KM*

1" by putting the reality condition onto the complex matrix Z i.e. z • if Z = Z

+ C3> where Z

+ denotes a hermitean conjugated matrix. A point in the twistor construction is the use of isomorphism between complex two dimensional matrices Z and Z—plane in a fourdimensional complex vector space C

4"— the twistor space ¥=C

4'. This isomporphism is given by the following correspondence L51 : Z * |subspace spanned by columns of 4x2 matrix [j^j^ or more explicitly, the 4x2 matrix columns are identified wit.h two twistors T ,T e IT : riz°+iz

3 z

2+iz

1 M 1 0 0 1 = CT

1,T

2) C4a> From a mathematical point of view the correspondence C43» gives an affine system of coordinates for the Z—plane in the twistor space "IT. This subspace is a complex Grassmann manifold G

2 ^CO. In other words, the Z—plane is given by the two linearly independent twistors T

±,T «= ¥. Therefore, the relation C4) gives us the correspondence between the complexified spacetime point z <£ CM

4 and a complex Z—plane in the twistor space IT. On the other hand, there is not a unique relation between the pair of twistors CT

±,T 5 and the Z—plane generated by this pair. It is clear, that every pair of twistors is related to a nonsingular 2x2 matrix as follows. CT',Tp = CT

t/r

2)M CS> gives the same Z—plane in the twistor space IT.