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IV. Spin Connection
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:
- substitute expression (4.8) into definition (4.4) of the spin connection
bivector;
- replace ${{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \to {e_\mu }^a{e_\nu
}^b{{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }$;
- use expression (1.29) for the Levi-Civita connection coefficients.
This then yields the useful relation
\[ 2{{\mathbf{\omega }}_\mu } =
{{\mathbf{g}}^\nu } \wedge \nabla {g_{\mu \nu }} + {{\mathbf{g}}_\nu } \wedge
{\partial _\mu }{{\mathbf{g}}^\nu }{\quad}{{\mathbf{g}}^\nu } = {e^\nu
}_b{{\mathbf{e}}^b} \]
first derived by John Snygg. 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 $)
\[ {g_{\mu \mu}} = {\eta _\mu
}{({e_\mu }^\mu )^2}{\quad} {\eta _\mu }: = {\eta _{\mu \mu }} \]
with ${\eta _\mu } = (1, - 1, - 1, - 1)$ the signature indicator. The
second term in relation (4.10) then vanishes
\[ {{\mathbf{g}}_\nu } \wedge {\partial
_\mu }{{\mathbf{g}}^\nu } = {{\mathbf{g}}_\nu } \wedge ({\partial _\mu }{e^\nu
}_\nu ){{\mathbf{e}}^\nu } = {g_{\nu\nu} }{e^\nu }_\nu ({\partial _\mu }{e^\nu
}_\nu ){{\mathbf{e}}^\nu } \wedge {{\mathbf{e}}^\nu } = 0 \]
and equation (4.10) reduces for diagonal metrics to
\[ 2{{\mathbf{\omega }}_\mu } =
{{\mathbf{g}}^\mu } \wedge \nabla {g_{\mu \mu }} = {{\mathbf{g}}^\mu } \wedge
{{\mathbf{g}}^\nu }{\partial _\nu }{g_{\mu \mu }} \]
without sum over $\mu $. This formula will be applied to the Schwarzschild
metric in Section VIII.