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

Tetrads in General Relativity

IV. Spin Connection

Snygg Relation

   In the torsionless case, it is possible to derive an expression for the spin connection bivector (4.4) in terms of the metric and the vierbein. The steps involved are:

  1. substitute expression (4.8) into definition (4.4) of the spin connection bivector;
  2. replace ${{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \to {e_\mu }^a{e_\nu }^b{{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }$;
  3. use expression (1.29) for the Levi-Civita connection coefficients.

This then yields the useful relation

4.10

\[  2{{\mathbf{\omega }}_\mu } = {{\mathbf{g}}^\nu } \wedge \nabla {g_{\mu \nu }} + {{\mathbf{g}}_\nu } \wedge {\partial _\mu }{{\mathbf{g}}^\nu }{\quad}{{\mathbf{g}}^\nu } = {e^\nu }_b{{\mathbf{e}}^b} \]

first derived by John Snygg [JS]. Note that, in the calculation of connection coefficients, the rule is that spacetime derivatives do not act on base vectors, only on scalar components relative to base vectors; see GA Coderivative, item a.

   A simplification occurs in the case of a diagonal metric: ${g_{\mu \mu }} = {\text{(}}{g_{00}},{g_{11}},{g_{22}},{g_{33}})$. If then the tetrad frame is aligned with the coordinate frame, the vierbein is also diagonal (no summation of $\mu $)

4.11

\[  {g_{\mu \mu}} = {\eta _\mu }{({e_\mu }^\mu )^2}{\quad} {\eta _\mu }: = {\eta _{\mu \mu }}  \]

with ${\eta _\mu } = (1, - 1, - 1, - 1)$ the signature indicator. The second term in relation (4.10) then vanishes

4.12

\[ {{\mathbf{g}}_\nu } \wedge {\partial _\mu }{{\mathbf{g}}^\nu } = {{\mathbf{g}}_\nu } \wedge ({\partial _\mu }{e^\nu }_\nu ){{\mathbf{e}}^\nu } = {g_{\nu\nu} }{e^\nu }_\nu ({\partial _\mu }{e^\nu }_\nu ){{\mathbf{e}}^\nu } \wedge {{\mathbf{e}}^\nu } = 0  \]

and equation (4.10) reduces for diagonal metrics to

4.13

\[  2{{\mathbf{\omega }}_\mu } = {{\mathbf{g}}^\mu } \wedge \nabla {g_{\mu \mu }} = {{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }{\partial _\nu }{g_{\mu \mu }} \]