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

Tetrads in General Relativity

IV. Spin Connection

Structure Coefficients

   The first term at the left-hand side of (4.27) is the exterior derivative

4.28

\[{\text{d}}{{\mathbf{e}}^a} = ({\partial _{[\mu }}{e_{\nu ]}}^a){{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu } : = \frac{1}{2}{f}_{\mu \nu }{^a}{{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }\]

defining the so-called structure coefficients as curls of the vierbeins. When expressed in the tetrad basis

4.29

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

they are known as coefficients (or objects) of anholonomy because they specify how much the tetrad frame $\{ {{\mathbf{e}}_a}\} $ departs from being holonomic, i.e., ${\partial _{[\mu }}{e_{\nu ]}}^c = 0$, as in the case of the coordinate basis $\{ {{\mathbf{g}}_\mu }\} $ being a gradient basis; see (1.25).

   By insertion of (4.25) and (4.28) into the torsionless Cartan equation (4.27), it is seen that this equation is satisfied only if:

4.30

\[2{\omega _{[ab]c}} =  {f_{abc}} \]

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:

4.31

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

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