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VIII. Schwarzschild solution
In terms
of vierbeins, the line element of GRT has the general form, see (2.3):
\[ d{s^2} = {g_{\mu \nu }}d{x^\mu
}d{x^\nu } = {\eta _{ab}}{e_\mu }^a{e_\nu }^bd{x^\mu }d{x^\nu } \]
The Schwarzschild metric (8.1) is diagonal and it is then a simple
matter to identify the corresponding vierbeins:
\[ \begin{gathered} &{e_\mu
}^a = {\text{diag[}}{e^\Phi }, {e^\lambda },r,r\sin \theta ]\\ &{e^\mu }_a
= {\text{diag[}}{e^{ - \Phi }},{e^{ - \lambda }},{r^{ - 1}}, {r^{ - 1}}{(\sin
\theta )^{ - 1}}] \end{gathered} \]
In
general, the vierbein fields $\{{e_\mu }^a(x)\}$ define the local
tetrads $\{ {{\mathbf{e}}_a}(x)\} $ through (2.5),(2.6). However, in a diagonal
metric, the coordinate basis ${{\mathbf{g}}_\mu }$ will be orthogonal.
The tetrad base may then be aligned with this coordinate base:
${{\mathbf{e}}_a} = {\eta _{ab}}{e_\mu }^b\nabla {x^\mu }$, with $\{ {x^\mu
}\} : = (t,r,\theta ,\phi )$ the Schwarzschild coordinates. Normalization to
the Minkowski metric gives:
\[ \begin{gathered} &{g_{tt}}
:= {{\mathbf{g}}_t} \cdot {{\mathbf{g}}_t}{\text{ = }}{{\text{e}}^{2\Phi }}
\qquad &{{\mathbf{e}}_t} = {e^{ - \Phi }}{{\mathbf{g}}_t} \hfill \\
&{g_{rr}}: = {{\mathbf{g}}_r} \cdot {{\mathbf{g}}_r}{\text{ = }} -
{e^{2\lambda }}\quad &{{\mathbf{e}}_r} = {e^{ - \lambda }}{{\mathbf{g}}_r}
\hfill \\ &{g_{\theta \theta }} := {{\mathbf{g}}_\theta } \cdot
{{\mathbf{g}}_\theta }{\text{ = }} - {r^2}\quad &{{\mathbf{e}}_\theta } =
{r^{ - 1}}{{\mathbf{g}}_\theta } \hfill \\ &{g_{\phi \phi }}: =
{{\mathbf{g}}_\phi } \cdot {{\mathbf{g}}_\phi }{\text{ = }} -
{r^2}{\text{si}}{{\text{n}}^2}\theta \qquad &{{\mathbf{e}}_\phi } =
{(r\sin \theta )^{ - 1}}{{\mathbf{g}}_\phi } \hfill \\ \end{gathered} \]
This choice of tetrads very much simplifies further calculations.