Spinor connections in general relativity

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Abstract
An axiomatic foundation is provided for the theory of spinor calculus on the space‐time manifold of general relativity. The methods, which deal directly with concepts in a coordinate free manner, allow not only elegant and compact statements of definitions and formulas but also have served as powerful analytical tools for the derivation of some interesting new results and for the unification and clarification of the previous work of other authors. The most general spinor connections are defined and related to the standard spinor connection (the unique spinor connection which is compatible with the spinor inner product and generates the Riemann 4‐vector connection). By means of the intrinsic formalism presented here an interpretation is given to the spinor theory of Infeld and van der Waerden. The most general spinor curvature tensors are derived, and two alternate expressions result from two bispinor connections which satisfy the desirable requirement of producing the standard 4‐vector connection by two different prescriptions. Another application of the techniques developed here results in an interesting expression for the spinor connection coefficients in terms of Dirac gamma matrices for four component spinors and arbitrary spinor connections which is much simpler and more general than others given in the literature.

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