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VIII. Schwarzschild Solution
In the GA formalism, the connection coefficients for the
Schwarzschild metric (8.1) can be obtained via the corresponding
connection bivectors (4.5). In this particular orthogonal case, the
formula to use is (4.14.). Note that from the definition of the
gradient $\nabla : = {{\mathbf{g}}^\mu }{\partial _\mu }$, it
follows that $\nabla t = {{\mathbf{g}}^\mu }{\partial _\mu }t =
{{\mathbf{g}}^t}$ and $\nabla r = {{\mathbf{g}}^\mu }{\partial _\mu
}r = {{\mathbf{g}}^r}$. Since the components of the Schwarzschild
metric tensor only depend on the radius $r$ :
\[ \begin{gathered}
&\nabla : = {{\mathbf{g}}^r}{\partial _r} = {e^\Phi
}{{\mathbf{e}}^r}{\partial _r}\\ &{{\mathbf{e}}_r} = {e^{ -
\lambda }}{{\mathbf{g}}_r} = {e^{ - \lambda
}}{g_{rr}}{{\mathbf{g}}^r} = - {e^\lambda }{{\mathbf{g}}^r}
\end{gathered} \]
With the above rules, the connection coefficients for the
Schwarzschild solution can directly be calculated from the metric:
\[ \begin{gathered}
&{{\mathbf{\omega }}_t} = - ({\partial _r}\Phi ){e^{\Phi -
\lambda }}{{\mathbf{e}}_t} \wedge
{{\mathbf{e}}_r}{\quad}&{{\mathbf{\omega }}_\theta } = {e^{ -
\lambda }} {{\mathbf{e}}_r} \wedge {{\mathbf{e}}_\theta } \hfill \\
&{{\mathbf{\omega }}_r} = - ({\partial _t}\lambda ){e^{\lambda -
\Phi }} {{\mathbf{e}}_t} \wedge
{{\mathbf{e}}_r}{\quad}&{{\mathbf{\omega }}_\phi } = \left(
{{e^{ - \lambda }}\sin \theta {{\mathbf{e}}_r} + \cos \theta
{{\mathbf{e}}_\theta }} \right) \wedge {{\mathbf{e}}_\phi }
\end{gathered} \]
Since the coefficients in the above connections only depend on the
radius, it follows that ${e^\lambda }{\partial _t}\lambda =
{\partial _t}{e^\lambda } = 0$ and
\[ {e^\Phi }{\partial _r}\Phi
= \frac{1}{{\sqrt {1 - 2M/r} }}\frac{M}{{{r^2}}} : = g
\]
Hence, the Schwarzschild connections are determined by three
bivectors with three connection coefficients. It
should be noted that the wedges $ \wedge $ in (8.10) are actually
unnecessary because the base vectors are orthogonal.