Expansion tensor in the metrics of Einstein and Schwarzschild

Dragi Radojević

Theoretical and applied mechanics, Tome 10 (1984) no. 1, p. 113 .

Résumé

In this paper we investigate the expansion tensor in the cosmological models of Einstein and Schwarzschild. We shall consider the exspansion tensor defined as \[ ǎrtheta_{lpha\beta}=\mathcal L_\xi(g_{lpha\beta}+\xi_lpha\xi_\beta) \] where $\xi_\alpha$ is the unit time-like vector pointed into the future, tangent on the world lines, $g_{\alpha\beta}$ is the metric tensor of the considered space, and $\mathcal L_\xi$ denotes Lie derivative with respect to $\xi^\alpha$. In relation to the field of radial four-speeds (observers), it turns out that the expansion is positive in both metrics. In the Schwarzschild metric, coordinate transformations of the observer are determined in a linear approximation.