By P. W Bridgman

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19) The flow τ generated by L defines a diffeomorphism of Su,u onto Su+τ,u , while the flow τ generated by L defines a diffeomorphism of Su,u onto Su,u+τ . The positive function defined by (20) g(L, L) = −2 2 may be thought of as the inverse density of the double null foliation. We denote by Lˆ and Lˆ the corresponding normalized future-directed null vectorfields Lˆ = −1 L, Lˆ = −1 ˆ L) ˆ = −2. L, so that g( L, (21) The first step is the analysis of the equations along the initial hypersurface Cu 0 .

The reason why this is required is that although the Gauss equation gives us L ∞ control on K , the estimate is not suitable for our purposes because it involves the loss of a factor of δ 1/2 in behavior with respect to δ. Thus one can only rely on the estimate obtained by integrating a propagation equation, which although optimal from the point of view of behavior with respect to δ, only gives us L 2 control on K . 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 .

The inner product g(L , L ) is then a negative function on M, hence there is a positive function on M defined by −g(L , L ) = 2 −2 . 10) We then define the normalized null vectorfields Lˆ and Lˆ by L , Lˆ = Lˆ = L. 11) These shall be used at each point in M \ 0 as a basis for the orthogonal complement of the tangent plane to the surface Su,u through that point. We have ˆ L) ˆ = −2. 12) We proceed to define the null vectorfields L and L by L= Lˆ = 2 Lˆ = L, L= 2 L. 10) we have: Lu = 0, Lu = 1, Lu = 1, Lu = 0.

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