Since twistor theory enables one to reconstruct the space-time geometry out of conformally invariant geometrical objects, it is important to know the basic tools for studying conformal gravity within the framework of general relativity. This is achieved by defining and using the Bach and Eastwood-Dighton tensors, here presented in two-spinor form (relying on previous work by Kozameh, Newman and Tod). After defining C-spaces and Einstein spaces, it is shown that a space-time is conformal to an Einstein space if and only if some equations involving the Weyl spinor, its covariant derivatives, and the trace-free part of Ricci are satisfied. Such a result is then extended to complex Einstein spaces. The conformal structure of infinity of Minkowski space-time is introduced in the end.
KeywordsBianchi Identity Einstein Space Twistor Theory Weyl Spinor Conformal Gravity
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