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.