Fermi Walker transport vs Thomas precession

[Jim Kata]

I can not derive the Fermi Walker transport equation to save my life, help! The Fermi Walker transport equation is [math]

\frac{{d\hat e_\alpha }}

{{d\tau }} = - \Omega \bullet \hat e_a

[/math] where [math]

\Omega ^{\mu \nu } = u^\mu a^\nu - a^\mu u^\nu

\[/math] with u and a being the proper velocity and acceleration. I can derive the equation for a Thomas precession with

[math]

A_T = I + \frac{{\gamma ^2 }}

{{\gamma + 1}}(\vec v \times \delta \vec v)S + (\gamma ^2 \delta \vec v_\parallel + \gamma \delta \vec v_ \bot )K

\[/math]

with [math]

S_{1 = } \left( {\begin{array}{*{20}c}

0 & 0 & 0 & 0 \\

0 & 0 & 0 & 0 \\

0 & 0 & 0 & { - 1} \\

0 & 0 & 0 & 1 \\

 

\end{array} } \right)

\[/math]

,c=1 and so on. I'm drunk and tired, but looking at these equations the Fermi Walker transport looks similar to Thomas precession except for a factor of [math]

\frac{{\gamma ^2 }}

{{\gamma + 1}}

\[/math] and some other differences. What is the connection between these formulas, and how do I derive the formula for the Fermi Walker transport? Please do not quote Misner chapter 6, chapter 8, or chapter 13.