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IX. Spherically Symmetric Spacetimes
The calculation of the connection bivectors starts with the
identification of metric tensor component from (9.2) and the
relation between the coordinate basis and the tetrad base from
(9.1):
\[ \begin{gathered}
&{g_{tt}} = {{\mathbf{g}}_t} \cdot {{\mathbf{g}}_t} =
{\text{g}}_1^2 - {\text{g}}_2^2 {\quad}&{{\mathbf{g}}_t} =
{g_1}{{\mathbf{e}}_t} - {g_2}{{\mathbf{e}}_r}{\quad} \hfill \\
&{g_{rr}} = {{\mathbf{g}}_r} \cdot {{\mathbf{g}}_r} = f_2^2 -
f_1^2{\quad}&{{\mathbf{g}}_r} = {f_1}{{\mathbf{e}}_r} -
{f_2}{{\mathbf{e}}_t} \quad \hfill \\ &{g_{\theta \theta }} =
{{\mathbf{g}}_\theta } \cdot {{\mathbf{g}}_\theta }{\text{ = }} -
{r^2}{\qquad} &{{\mathbf{g}}_\theta } = r{{\mathbf{e}}_\theta }
\qquad \qquad \hfill \\ &{g_{\phi \phi }} = {{\mathbf{g}}_\phi }
\cdot {{\mathbf{g}}_\phi }{\text{ = }} - {r^2}{\sin}^2\theta
{\quad}&{{\mathbf{g}}_\phi } = r\sin \theta {{\mathbf{e}}_\phi }
\qquad \hfill \\ &{g_{tr}} = {{\mathbf{g}}_t} \cdot
{{\mathbf{g}}_r} = {f_1}{g_2} - {g_1}{f_2} \quad \hfill \\
\end{gathered} \]
These relations imply that the tetrad base $\left\{
{{{\mathbf{e}}_t},{{\mathbf{e}}_r},{{\mathbf{e}}_\theta
},{{\mathbf{e}}_\phi }} \right\}$ is orthonormal.
The connection coefficients may then be obtained by way of
the Snygg formula (4.14):
\[ \begin{gathered}
{{\mathbf{\omega }}_t} = - ({g_1}{\partial _r}{g_1} - {g_2}{\partial
_r}{g_2}){{\mathbf{e}}_t} \wedge {{\mathbf{e}}_r}{\text{ }} \hfill
\\ {{\mathbf{\omega }}_r} = - ({f_1}{\partial _r}{g_2} -
{f_2}{\partial _r}{g_1}){{\mathbf{e}}_t} \wedge
{{\mathbf{e}}_r}{\text{ }} \hfill \\ {{\mathbf{\omega }}_\theta } =
{g_1}{{\mathbf{e}}_r} \wedge {{\mathbf{e}}_\theta } -
{g_2}{{\mathbf{e}}_t} \wedge {{\mathbf{e}}_\theta } \quad \qquad
\hfill \\ {{\mathbf{\omega }}_\phi } = \sin \theta \left(
{{g_1}{{\mathbf{e}}_r} - {g_2}{{\mathbf{e}}_t}} \right) \wedge
{{\mathbf{e}}_\phi } + \cos \theta {{\mathbf{e}}_\theta } \wedge
{{\mathbf{e}}_\phi } \hfill \\ \end{gathered} \]
No derivatives of ${f_1},{f_2}$ appear.
A comparison of (9.2) with (8.1) shows that the Schwarzschild
metric corresponds to the parameter choice
\[ {f_1} =
\frac{1}{{{g_1}}}{\quad}{f_2} = 0{\quad}g_1^2 = 1 - \frac{{2M}}{r} =
{e^{2\Phi }}{\quad}{g_2} = 0 \]
Substituting these parameters values into (9.4), one recovers the
Schwarzschild connection coefficients (8.10).
Note that ${f_1}{g_2} - {f_2}{g_1} = 1$ in this case. However, there
are other possible choices; for example, $f_1^2 - f_2^2 = 0$ leads
to the Eddington-Finkelstein metric, and $f_1^2 - f_2^2 = 0$,
to the Painlevé-Gullstrand metric.