Most of the detailed equations that I give here take advantage of the 2-spinor formalism. This is not a matter of necessity, since the more familiar 4-tensor description could have been provided pretty well throughout, as an alternative. However, not only is the 2-spinor formalism simpler when it comes to expressing conformal invariance properties (see A6), but it also provides a more systematic overview when it comes to understanding the propagation of massless fields and the corresponding Schrodinger equation for their constituent particles.
Conventions employed here, including the use of abstract indices, are as in Penrose and Rindler (1984, 1986),[A1] except that A denotes the cosmological constant here, rather than the 'A' of that work, and the scalar curvature quantity 'A' that appears there would bej^R. References to equations starting with 'P&R' refer to that work, and in fact all the needed equations are to be found in the 1986 Volume 2. The Einstein tensor Eab used here is the negative of the 'Einstein tensor' Rab —§Rgab used there (with the same sign of Ricci tensor Rab as adopted there), so that the Einstein field equations become (as in §2.6 and §3.5)
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