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Atman

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**Derive "Focusing Theorem" in Gravitation by MTW**

I followed the logic in Chapter 22.5 from Misner&Thorne&Wheeler, trying to derive the focusing theorem, i.e. equation(22.37).

The problem that blocks me is the derivation of (22.36) for the cross section area. I already find the equation for evolution of the vector connecting adjacent light rays on the hypersurface. The thing is, the authors don't even give an explicit mathematical expression for the cross section area. And if I define the area square as (V^2 *W^2 - (V*W)^2 ), the RHS of(22.36) does come out easily, since it is a contraction of k, the tangent vector of the null geodesic, to Nabla, while the Nabla also acting on k.

For those who have succeeded in this, is there any trick that makes such a contraction possible? Working on 2-D submanifold? Or anyone who are more expert in these stuff, could you give any suggestion? It really puzzles me for a long time. Thank you~