By Wolfgang Rindler

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What makes possible the gain of one degree 26 Prologue of differentiability by considering systems of ordinary differential equations along the generators of the Cu or the C u coupled to elliptic systems on the Su,u sections, is the fact that the principal terms in the propagation equations for certain optical entities vanish, by virtue of the Einstein equations. In the case of the coupled system pertaining to χ and χ, these entities are simply the traces trχ and trχ, and the Codazzi equations constitute the elliptic systems for the trace-free parts χˆ and χˆ respectively.

56) χ × χ − α. 54) takes the form Dχ = 2 Let α be the spacetime curvature component which is the symmetric 2-covariant S tensorfield given by ˆ Y, L) ˆ : for any pair X, Y ∈ T p Su,u . 58) Proceeding in an analogous manner, with C u , L, χ in the roles of Cu , L, χ respectively, we deduce the equation Dχ = ωχ + (χ × χ − α). 2. The optical structure equations 41 we have −1 χ = χ. 61) χ × χ − α. 62), as the second variational formulas. 62) are propagation equations along the generators of each C u .

The set of quantities {q1 , . . , qn } is gradually enlarged as we proceed through the sequence of chapters. 1, which occupies the four succeeding chapters. As we mention above, Chapter 10 represents a new beginning. The point is the following. Chapters 3–7 derive estimates for the connection coefficients in terms of quantities involving the L ∞ norms on the Su,u of the curvature components, the L 4 norms on the Su,u of the 1st derivatives of the curvature components, and the L 2 norms on the Cu of the 2nd derivatives of the curvature components (with the exception of those involving α and the 2nd transversal derivatives of β).