THE KERR THEOREM, MULTISHEETED TWISTOR SPACES AND MULTIPARTICLE KERR-SCHILD SOLUTIONS

Abstract

multiparticle solutions. 1.Introduction. The Kerr-Newman solution displays many relationships to the quantum world. It is the anomalous gyromagnetic ratio g = 2, stringy structures and other features allowing one to construct a semiclassical model of the extended electron 1–3 which has the Compton size and possesses the wave properties. One of the mysteries of the Kerr geometry is the existence of two sheets of space-time, (+) and (−), on which the dissimilar gravitation (and electromagnetic) fields are realized, and fields living on the (+)-sheet do not feel the fields of the (−)sheet. Origin of this twofoldedness lies in the Kerr theorem, generating function F of which for the Kerr-Newman solution has two roots which determine two different twistorial structures on the same space-time. The standard Kerr-Schild formalism is based on a restricted version of the Kerr theorem which uses polynomials of second degree in Y, and, in fact, produced only the Kerr geometry. The use of Kerr theorem in full power is related with the treatments of polynomials of higher degrees in Y. On this way we obtain the multisheeted twistor spaces and corresponding multiparticle Kerr-Schild solutions. 4,5 The case of a quadratic in Y generating function of the Kerr Theorem F(Y ) was investigated in details in. 6,7 It leads to the Kerr spinning particle (or black hole) with an arbitrary position, orientation and boost. Choosing generating function F(Y ) as a product of partial functions Fi for spinning particles i=1,...k, we obtain multi-sheeted, multitwistorial space-time over M 4 possessing unusual properties. Twistorial structures of the i-th and j-th particles turn out to be independent, forming a type of its internal space. However, the exact solutions show that gravitation and electromagnetic interaction of the particles occurs via the connecting them singular twistor lines. The space-time of the multiparticle solutions turns out to be covered by a net of twistor lines, and we conjecture that it reflects its relation to quantum gravity. Recall that the Kerr-Newman metric can be represented in the Kerr-Schild form gµ� = �µ� + 2hkµk�, where �µ� is metric of auxiliary Minkowski space-time, and h = (mr −e 2 /2)/(r 2 +a 2 cos 2 �). kµ(x) is a twisting null field, which is tangent to the Kerr principal null congruence (PNC) which is geodesic and shear-free. 7–9