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Tetrads in General Relativity

IX. Spherically Symmetric Spacetimes

General Line Element

   The Schwarzschild solution may give the simplest form of the metric, but it does not offer the simplest or most physically transparent interpretation. For one, it is ill defined at the Schwarzschild radius because the coordinate system is singular there. However, for the case of stationary, spherically-symmetric spacetimes, there are exact solutions of the Einstein equations other than Schwarzschild’s.

   It is relevant to note here that the stationarity condition is not the same as the static condition, which is the statement that ${{\mathbf{g}}_t} \cdot {{\mathbf{g}}_r} = 0$. The static condition is highly restrictive, and leads automatically to the Schwarzschild metric. Stationarity, on the other hand, amounts to independence of all quantities of the time variable $t$. 

   By an analysis of Einstein’s vacuum (source-free) equations, it has been shown that the vierbeins associated with stationary, spherically-symmetric spacetimes, must be of the general form

9.1

\[ [{e_\mu }^a] = \left[ {\begin{array}{*{20}{c}} {{g_1}}&{ - {g_2}}&0&0 \\ { - {f_2}}&{{f_1}}&0&0 \\ 0&0&r&0 \\ 0&0&0&{r\sin \theta } \end{array}} \right]{\quad}[{e^\mu }_a] = \left[ {\begin{array}{*{20}{c}} {{f_1}}&{{f_2}}&0&0 \\ {{g_2}}&{{g_1}}&0&0 \\ 0&0&{{r^{ - 1}}}&0 \\ 0&0&0&{{{(r\sin \theta )}^{ - 1}}} \end{array}} \right] \]

where $f,g$ are functions of the radius $r$ only; the determinant of the non-diagonal bloc has the value ${f_1}{g_1} - {f_2}{g_2} = 1$; this last condition guarantees that the frames defined by (9.1) satisfy the torsion-free condition (1.26).

   By substitution of (9.1) in (8.6) one obtains the general spherically symmetric line element

9.2

\[  \begin{gathered} d{s^2}: = {{\mathbf{g}}_\mu } \cdot {{\mathbf{g}}_\nu }d{x^\mu }d{x^\nu } \hfill\\ = (g_1^2 - g_2^2)d{t^2} + 2({f_1}{g_2} - {f_2}{g_1})dtdr - (f_1^2 - f_2^2)d{r^2} - {r^2}d{\Omega ^2} \end{gathered}  \]

Here $d{\Omega ^2}: = (d{\theta ^2} + {\sin ^2}\theta d{\phi ^2})$ is an abbreviation for the metric of the unit sphere.