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By Demetrios Christodoulou

ISBN-10: 303719068X

ISBN-13: 9783037190685

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The Oi are then first defined on Cu 0 by conjugation with the flow of L and then in spacetime by conjugation with the flow of L. The Prologue 23 Weyl fields which we consider are, besides the fundamental Weyl field R, the Riemann curvature tensor, the following derived Weyl fields: 1st-order: L˜ L R, L˜ Oi R : i = 1, 2, 3, L˜ S R. 2nd-order: L˜ L L˜ L R, L˜ Oi L˜ L R : i = 1, 2, 3, L˜ O j L˜ Oi R : i, j = 1, 2, 3, L˜ Oi L˜ S R : i = 1, 2, 3, L˜ S L˜ S R. (35) We assign to each Weyl field the index l according to the number of L˜ L operators in the definition of W in terms of R.

Then L q of θ as a type T p tensorfield on Su,u with respect to X as a vectorfield on Su,u , a notion intrinsic to Su,u , which makes no reference to the ambient spacetime. 19) 36 Chapter 1. The Optical Structure Equations with X an arbitrary S vectorfield, holds, and so do the Leibniz rules D(η · θ ) = (Dη) · θ + η · (Dθ ), D(η · θ ) = (Dη) · θ + η · (Dθ ). 20) Let us recall that if X and Y are arbitrary vectorfields and θ an arbitrary tensorfield on any manifold, we have L X LY θ − LY L X θ = L[X,Y ] θ.

Thus L / X W is defined to be simply L X W , which is itself a q contravariant S tensorfield. Consider finally the general case of an arbitrary type q / X θ is again defined on each Su,u as the usual Lie derivative T p S tensorfield θ . Then L q of θ as a type T p tensorfield on Su,u with respect to X as a vectorfield on Su,u , a notion intrinsic to Su,u , which makes no reference to the ambient spacetime. 19) 36 Chapter 1. The Optical Structure Equations with X an arbitrary S vectorfield, holds, and so do the Leibniz rules D(η · θ ) = (Dη) · θ + η · (Dθ ), D(η · θ ) = (Dη) · θ + η · (Dθ ).

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The formation of black holes in general relativity by Demetrios Christodoulou


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