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Tetrads in General Relativity

IV. Spin Connection

Ricci Rotation Coefficients

   By defining $\,{\partial _a}: = {e^\mu }_a{\partial _\mu }$, $\,{D_a}: = {e^\mu }_a{D_\mu }$, $\,{{\bs{\omega }}_a}: = {e^\mu }_a{{\bs{\omega }}_\mu }$, one may write the definition of the spin connection (4.1) in the alternative forms

4.19

\[  \begin{gathered} {D_a}{{\mathbf{e}}_b} = {e^\nu }_a \Sigma {{_\nu}{^c}}{_b} {{\mathbf{e}}_c}  = {\omega{{_a}{^c}}{_b}} {{\mathbf{e}}_c}  = {{\bs{\omega }}_a} \cdot {{\mathbf{e}}_a}\\ {D_a}{{\mathbf{e}}^b} = - \omega {{_a}{^b}}{_c}{{\mathbf{e}}^c}  = - {{\bs{\omega }}_a} \cdot {{\mathbf{e}}^b}\end{gathered}  \]

   The tetrad connection coefficients

4.20

\[  {\omega _{abc}} : = {e^\nu }_a{\Sigma _{\nu bc}} \]

are called Ricci connection coefficients or Ricci rotation coefficients. They measure the relative rotation of frame tetrads when moved in various directions, encoding thus gravitational and non-inertial effects. Equivalently, from (4.6), (4.9), (4.19), the Ricci rotation coefficients may be defined by: [Wikipedia: Christoffel symbols]

4.21

\[  {\omega _{abc}}: = {{\mathbf{e}}_b} \cdot ({D_a}{{\mathbf{e}}_c}) = {{\mathbf{e}}_b} \cdot {{\bs{\omega }}_a} \cdot {{\mathbf{e}}_c} = {e^\mu }_a{e_{\nu [b}}D_\mu ^\Gamma {e^\nu }_{c]}  \]

   In addition to the spin connection bivector (4.5), by the Ricci coefficients one also defines a spin-connection 1-form

4.22

\[  {{\bs{\omega }}^a}_b: = {{\mathbf{g}}^\mu }\Sigma {{_\mu}{^a}}{_b} = {{\mathbf{e}}^c}{\omega{{_c}{^a}}{_b}} {\quad}{{\bs{\omega }}_{ab}} = - {{\bs{\omega }}_{ba}}  \]

in the mathematics of Lie groups known as the Maurer-Cartan connection. Its relation with the connection bivector (4.5) is through (4.6):

4.23

\[  {{\mathbf{g}}^\mu }{D_\mu }{{\mathbf{e}}_a}: = D{{\mathbf{e}}_a} = {{\mathbf{g}}^\mu }{{\bs{\omega }}_\mu } \cdot {{\mathbf{e}}_a} = {{\bs{\omega }}^b}_a{{\mathbf{e}}_b}  \]

   In spite of their close correspondence, ${{\bs{\omega }}_{ab}}$ and ${{\bs{\omega }}_\mu }$ are different geometric objects: the former is a matrix valued 1-form with values in the Lie-algebra of the Lorentz group $\text{SO}(1,3)$, whereas the latter is a bivector. The Maurer-Cartan connection plays a central role in the Cartan formalism of GRT defining both torsion and curvature. The connection bivector is the primary connection form in the GA tetrad-formalism.