# Lösung der Feldgleichungen

• Otto Heckmann

## Zusammenfassung

Berechnet man für die Metrik (22) nach (5) die $$\Gamma {{_{x}^{\lambda }}_{\theta }}$$, so bekommt man
$$\left\{ {\begin{array}{*{20}{c}} {\Gamma _{{ij}}^{k} = \mathop{{\Gamma _{{ij}}^{k}}}\limits^{*} = \frac{1}{2}{{\gamma }^{{kl}}}\left( {\frac{{\partial {{\gamma }_{{il}}}}}{{\partial {{x}^{j}}}} + \frac{{\partial {{\gamma }_{{lj}}}}}{{\partial {{x}^{i}}}} - \frac{{\partial {{\gamma }_{{ij}}}}}{{\partial {{x}^{l}}}}} \right);} \hfill \\ {\Gamma _{{ij}}^{4} = R\dot{R}{{\gamma }_{{ij}}};{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \Gamma _{{i4}}^{k} = \frac{{\dot{R}}}{R}\delta _{i}^{k};{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \delta _{i}^{k} = \left\{ {\begin{array}{*{20}{c}} {0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\text{wenn }}i \ne {\mkern 1mu} k;{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} } \hfill \\ {1{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} i = k;} \hfill \\ \end{array} } \right.} \hfill \\ {\Gamma _{{44}}^{k} = \Gamma _{{4k}}^{4} = \Gamma _{{44}}^{4} = 0.} \hfill \\ \end{array} } \right.$$
(22a)

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