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Tetrads in General Relativity

IV. Spin Connection

Structure Coefficients

   Setting the torsion to zero in equation (4.24), one derives the constraint equation (4.27) in component form

4.28

\[0 = 2\left( {{\partial _{[\mu }}{e_{\nu ]}}^a + {\Sigma _{[\mu |}}{{^a}_b}{e_{|\nu ]}}^b} \right) = {f_{\mu \nu }}^a - 2{e_{[\mu }}^b{e_{\nu ]}}^c{\omega _{bc}}^a\]

as an relation between the Ricci connection (4.20) and the so-called structure coefficients defined as the curl of the vierbeins. When expressed in the tetrad basis

4.29

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

these are also 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). In this special case, in the absence of torsion, the Ricci connection is symmetric in its two first indices.

   With (4.29) it follows that equation (4.28) 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 connection  emerges as a pure combination of the anholonomy coefficients.:

4.31

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

The anti-symmetry of the Ricci connection in its last two indices is ensured by the anti-symmetry of the structure coefficients in their first two indices.

   Thus, the (torsionless) spin connection is completely determined by the vierbeins. One may note the similarity to the Christoffel formula (1.29) for Levi-Civita connection coefficients. This is not by coincidence because it is a theorem that there exists a unique metric torsion-free connection which is the one given by (1.29) or, given a triad form, (4.31).