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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 (4.13):
\[ \begin{gathered}
{{\mathbf{\omega }}_t} = - ({g_1}{\partial _r}{g_1} - {g_2}{\partial
_r}{g_2}){{\mathbf{e}}_t} \wedge {{\mathbf{e}}_r}{\text{ }} \hfill \\
{{\mathbf{\omega }}_r} = - ({f_1}{\partial _r}{g_2} - {f_2}{\partial
_r}{g_1}){{\mathbf{e}}_t} \wedge {{\mathbf{e}}_r}{\text{ }} \hfill \\
{{\mathbf{\omega }}_\theta } = {g_1}{{\mathbf{e}}_r} \wedge
{{\mathbf{e}}_\theta } - {g_2}{{\mathbf{e}}_t} \wedge {{\mathbf{e}}_\theta }
\quad \qquad \hfill \\ {{\mathbf{\omega }}_\phi } = \sin \theta \left(
{{g_1}{{\mathbf{e}}_r} - {g_2}{{\mathbf{e}}_t}} \right) \wedge
{{\mathbf{e}}_\phi } + \cos \theta {{\mathbf{e}}_\theta } \wedge
{{\mathbf{e}}_\phi } \hfill \\ \end{gathered} \]
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.