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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 (4.4). In this
particular orthogonal case, the formula to use is (4.13.). 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.