# The Formation of Black Holes in General Relativity: 4 by Demetrios Christodoulou By Demetrios Christodoulou

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Y p ) and to each of the vectorfields Y1 , . . 20). ) The second part of the lemma is established in an analogous manner. 29) and ω = D log , ω = D log . 30) The tangent hyperplane T p Cu to a given null hypersurface Cu at a point p ∈ Cu consists of all the vectors X at p which are orthogonal to Lˆ p : T p Cu = {X ∈ T p M : g(X, Lˆ p ) = 0}. 31) We have Lˆ p ∈ T p Cu , Lˆ p being orthogonal to itself. 32) and we have Lˆ p ∈ T p C u , Lˆ p being orthogonal to itself. Thus the induced metrics on Cu and C u are degenerate.

The L p elliptic theory on the Su,u is applied in Chapter 6, in the case p = 4, to the elliptic systems mentioned above, to obtain L 4 estimates for the 2nd derivatives of the connection coefficients on the surfaces Su,u . 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.

Form of Cu is intrinsic to Cu , the vectorfield Lˆ being tangential to Cu . We have χ(X, Y ) = χ(P X, PY ). 38) Thus χ is a symmetric 2-covariant S tensorfield. We may consider χ to be the 2nd fundamental form of the sections Su,u of Cu relative to Cu . The 2nd fundamental form χ of a given null hypersurface C u is a bilinear form in T p C u at each p ∈ C u defined as follows. Let X, Y ∈ T p C u . Then ˆ Y ). 39) The bilinear form χ is symmetric. For, if we extend X, Y to vectorfields along C u which ˆ = g(Y, L) ˆ = 0, are tangential to C u , we have, in view of the fact that g(X, L) ˆ ∇ X Y ) + g( L, ˆ ∇Y X) = −g( L, ˆ [X, Y ]) = 0 χ(X, Y ) − χ(Y, X) = −g( L, ˆ We note that the 2nd fundamental as [X, Y ] is also tangential to C u hence orthogonal to L.