\( \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\bean}{\begin{eqnarray*}} \newcommand{\eean}{\end{eqnarray*}} \newcommand{\la}{\label} \newcommand{\nn}{\nonumber} \newcommand{\half}{{\scriptstyle \frac{1}{2}}} \newcommand{\third}{{\scriptstyle \frac{1}{3}}} \newcommand{\bli}[2]{\begin{list}{#1}{\itemsep=0.0cm \topsep=0.0cm \partopsep=0.0cm #2}} \newcommand{\eli}{\end{list}} \newtheorem{problem}{Problem}[chapter] \newcommand{\bprob}{\begin{problem}} \newcommand{\eprob}{\end{problem}}\)

Tetrads in General Relativity

IV. Spin Connection

Structure Coefficients

   The first term at the left-hand side of (4.25) is the exterior derivative/curl

4.27

\[\nabla \wedge {{\mathbf{e}}^a} = ({\partial _\mu }{e_\nu }^a){{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu } := - \frac{1}{2}{f^a}_{bc}{{\mathbf{e}}^b} \wedge {{\mathbf{e}}^c}  \]

which defines the so-called structure constants (or structure coefficients) as curls of the vierbeins

4.28

\[  {f^c}_{ab} : = {e^\mu }_a{e^\nu }_b({\partial _\nu }{e_\mu }^c - {\partial _\mu }{e_\nu }^c) = - 2{e^\mu }_a{e^\nu }_b{\partial _{[\mu }}{e_{\nu ]}}^c  \]

They are also known as coefficients of anholonomy because they specify how much the tetrad frame $\{ {{\mathbf{e}}_a}\} $ departs from being holonomic, i.e.,$\,{f^c}_{ab} = 0$, as in case of the coordinate basis $\{ {{\mathbf{g}}_\mu }\} $; see (1.25). 

   By insertion of (4.23) and (4.27) into the Cartan equation (4.25), it is seen that this equation is satisfied only if:

4.29

\[2{\omega _{[ab]c}} = - {f_{cab}} \]

By adding the same equation with a cyclic permutation of the free indices $abc \to cab$ and then subtracting a further cyclic permutation, the Ricci rotation coefficients emerge as a pure combination of structure constants:

4.30

\[  {\omega _{abc}} = \frac{1}{2}\left( {{f_{abc}} + {f_{cba}} - {f_{bca}}} \right)  \]

   Thus, the (torsionless) spin connection is completely determined by the vierbeins. Note that the structure coefficients, like the spin connection, are anti-symmetric in the last two indices. Note also the similarity to the Christoffel formula (1.29) for Levi-Civita connection coefficients.