I don't have time to do the full calculation (it will be rather long!), but I'll indicate what I think are the steps that get you there:

We start with our congruence given by the normalized vector field $u^a$, $u_au^a=-1$

The covariant derivative of $u^a$ splits into a part parallel to the congruence and a part orthogonal to it: $$\nabla_au_b=-u_a\dot{u_b}+{\tilde{\nabla}}_au_b $$ Where the tilde derivative is defined by projecting orthogonal to $u^a$ $${\tilde{\nabla}}_au_b=h_a^ch_b^d\nabla_cu_d $$ $$h_a^b=\delta_a^b+u_au^b $$Now we can decompose ${\tilde{\nabla}}_au_b$ into its irreducible parts $$ {\tilde{\nabla}}_au_b = \omega_{ab}+\frac{1}{3}\Theta h_{ab}+\sigma_{ab}$$ Where $\omega_{ab}$ is the antisymmetric part, $\Theta$ is the trace part, and $\sigma_{ab}$ is the trace-free symmetric part.

In most derivations of the Gauss-Codazzi equations, they assume that $u_a$ is vorticity-free ($\omega$ is the vorticity). Here we can't make that assumption. We wish to investigate curvature orthogonal to the congruence so we want to calculate $$({\tilde{\nabla}}_a{\tilde{\nabla}}_b-{\tilde{\nabla}}_b{\tilde{\nabla}}_a)X_c $$ where X is a vector field orthogonal to the congruence. Directly substituting for the ${\tilde{\nabla}}$ factors a couple of pages of calculation got me to $$ ({\tilde{\nabla}}_a{\tilde{\nabla}}_b-{\tilde{\nabla}}_b{\tilde{\nabla}}_a)X_c$$ $$=2\omega_{ab}{\dot{X}}_{<c>}+(^{\perp}R_{abcd})X_d+(K_{cb}K_{da}-K_{ca}K_{db})X^d $$where$$K_{ab}={\tilde{\nabla}}_bu_a $$ (I'm using the angle brackets and time derivatives defined in eqns (9) and (10) of your reference and the perp just means project all free indices using the $h$'s. Also the Gauss Codazzi section in Wald is useful here. Oh BTW I can't guarantee signs and factors of two!).

I believe the next step would be to contact this equation to obtain the desired three-Ricci tensor. It contains all the ingredients in your desired equation (54). The only problem is that you still have the (projected) Riemann tensor involved. To get rid of that you would have to use the field equations - this will bring in ingredients like $\pi_{ab}$

Sorry - it's more of a long hint than an answer, but it is a rather messy calculation! (you may have already completed it yourself now...)

This post imported from StackExchange Physics at 2014-03-22 17:15 (UCT), posted by SE-user twistor59