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Tetrads in General Relativity

IV. Spin Connection

Tetrad Postulate

   A bit of manipulation allows to write the relations (4.7),(4.8) as the vanishing of the coderivative of the vierbeins

4.14

\[  {D_\mu }{e_\nu }^a = {\partial _\mu }{e_\nu }^a - \Gamma _{\mu \nu }^\lambda {e_\lambda }^a + {\Sigma {{_\mu}{^a}}{_b}}{e_\nu }^b = 0  \]

4.15

\[{D_\mu }{e^\nu }_a = {\partial _\mu }{e^\nu }_a + \Gamma _{\mu \lambda }^\nu {e^\lambda }_a - {\Sigma {{_\mu}{^b}}{_a}}{e^\nu }_b = 0\]

Either one of these equations is known as the tetrad postulate. However, they are just a restatement of the relation between the Levi-Civita and the spin connections and express the fact that the total covariant derivative of the tetrads with respect to both spacetime and tetrad indices is vanishing. It is only a postulate in the sense that geometric objects are supposed to be independent of their coordinates and basis elements.

   From the tetrad postulate immediately follows the metric compatibility of the metric tensor expressed as the vanishing of the coderivative:

4.16

\[  {D_\mu }{g_{\nu \lambda }} = {D_\mu }({e_\nu }^a{e_\lambda }^b{\eta _{ab}}) = ({D_\mu }{e_\nu }^a){e_\lambda }^b{\eta _{ab}} + {e_\nu }^a({D_\mu }{e_\lambda }^b){\eta _{ab}} = 0  \]

Thus, metric compatibility is equivalent to the anti-symmetry of the spin connection in its Latin indices. Stated differently, metric compatibility holds if and only if  the connection is Lorentzian.

   The tetrad postulate also implies that one can switch index types through the coderivative, e.g.

4.17

\[  {D_\mu }{A^a} = {D_\mu }({e_\nu }^a{A^\nu }) = {e_\nu }^a{D_\mu }{A^\nu }  \]

This important property is frequently used in calculations with tensor components.