Derive Focusing Theorem in Gravitation by MTW

[Atman]

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~

 

[Ambitwistor]

Thank you for your question regarding the derivation of the focusing theorem in gravitation from Misner, Thorne, and Wheeler's book. I understand your frustration with the lack of an explicit mathematical expression for the cross section area in equation (22.36). I will try to explain the steps to derive this equation and hopefully it will help clarify the process for you.

First, let's define some variables that will be used in the derivation. Let the null geodesic connecting two neighboring light rays be denoted by k and let the tangent vector to this geodesic be denoted by k^a. The cross section area will be denoted by A and the area element will be denoted by dA. We will also use the notation for the expansion parameter, Θ, and the shear tensor, σ_ab.

Now, the key idea in deriving equation (22.36) is to use the Raychaudhuri equation, which describes the evolution of the expansion parameter Θ along a null geodesic. This equation is given by:

dΘ/dλ = -1/2Θ^2 - σ_abσ^ab - R_abk^ak^b

where λ is an affine parameter along the geodesic and R_ab is the Ricci tensor. Using this equation, we can express the change in the cross section area as:

dA/dλ = AΘ + σ_abk^ak^bdA

Now, we need to relate the area element dA to the shear tensor σ_ab. This can be done by considering the infinitesimal area element on a 2-dimensional submanifold of our 4-dimensional spacetime. This submanifold is spanned by the two light rays and their tangent vectors. Using the fact that the infinitesimal area element on a 2-dimensional surface is given by dA = |k^a x l^a|dλ, where l^a is the other tangent vector on the submanifold, we can express dA in terms of the shear tensor as:

dA = √(σ_abσ^ab)dλ

Substituting this into our equation for dA/dλ, we get:

dA/dλ = AΘ + √(σ_abσ^ab)k^ak^bdλ

Now, we can use the Raychaudhuri equation again to express the Ricci tensor term in