Global Solutions of Einstein–dirac Equation

The conformal space M was introduced by Dirac in 1936. It is an algebraic manifold with a spin structure and possesses naturally an invariant Lorentz metric. By carefully studying the birational transformations of M, we obtain explicitly the transition functions of the spin bundle over M. Since the transition functions are closely related to the propagation in physics, we get a kind of solutions of the Dirac equation by integrals constructed from the propagation. Moreover, we prove that the invariant Lorentz metric together with one of such solutions satisfies the Einstein-Dirac combine equation. 1. The main results. In general relativity the 4-dimensional Lorentz manifold is used. It is Penrose [1] who began to apply 2-component spinor analysis for studying Einstein equation. It implied that the spin group Spin(1, 3) of a Lorentz spin manifold M is locally isomorphic to the group SL(2,C) such that there is a Lie group homeomorphism ι : SL(2,C) −→ SO(1, 3) which is a two to one covering map. Then a two component Dirac operator D : V2(x) → V ∗ 2 (x) and D : V ∗ 2 (x) → V2(x) can be defined, where V2(x) is the vector space of spinors at x ∈ M and V ∗ 2 (x) is the conjugate vector space of V2(x). We will use the following lemma for studying the Dirac equation. Lemma 1. If ψ is a two component spinor field on M and satisfies Dψ = DDψ = −m2ψ (1.1)