By Demetrios Christodoulou
ISBN-10: 303719068X
ISBN-13: 9783037190685
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Extra info for The formation of black holes in general relativity
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14) we have (du)(L) = 1, while (du)(Y ) = 0, the S 1-form L / X θ is given on the manifold Cu by: L / X θ = L X θ − f du where f = θ ([L, X]). 22) On the other hand, since (Dθ )(Y ) = (L L θ )(Y ) while (Dθ )(L) = 0 = (L L θ )(L), the S 1-form Dθ is given on the manifold Cu simply by: Dθ = L L θ. 23) Again, let Y be an arbitrary S vectorfield. 14). 23), (L / X Dθ )(Y ) = (L X Dθ )(Y ) = (L X L L θ )(Y ). 24) we then obtain / X Dθ )(Y ) = (L L L X θ − L X L L θ )(Y ) = (L[L ,X ] θ )(Y ) = (L / [L ,X ] θ )(Y ).
76) reads −Dζ = −d/ω + χ · ( ζ + d/ ) + β. 69) reads ∇ X Lˆ = ζ(X) Lˆ + χ · X. 30) we have d/ω = Dd/ log , d/ω = Dd/ log . 82) Dη = (χ · η + β). 83) is a propagation equation along the generators of each C u . The S 1-form η may be considered to be the torsion of Su,u with respect to the null geodesic field L , because η(X) = 2 2 g(∇ X L , L ) : ∀X ∈ T Su,u . 84) Similarly, the S 1-form η may be considered to be the torsion of Su,u with respect to the null geodesic field L , because η(X) = 2 2 g(∇ X L , L ) : ∀X ∈ T Su,u .
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. It then readily follows that if X and Y are S vectorfields and θ an S tensorfield of any type, we have: L / XL /Yθ − L /YL / Xθ = L / [X,Y ] θ. 21) Moreover, we have the following lemma.