Page:The Meaning of Relativity - Albert Einstein (1922).djvu/98

is so chosen that $g$ = constant. From $R_{\mu \nu }$ we can form the scalar,

$R=g^{\mu \nu }R_{\mu \nu }.$

(89)

Straightest (Geodetic) Lines. A line may be constructed in such a way that its successive elements arise from each other by parallel displacements. This is the natural generalization of the straight line of the Euclidean geometry. For such a line, we have

$\delta \left({\frac {dx_{\mu }}{ds}}\right)=-\Gamma _{\alpha \beta }^{\mu }{\frac {dx_{\alpha }}{ds}}dx_{\beta }.$

The left-hand side is to be replaced by ${\frac {d^{2}x_{\mu }}{ds^{2}}}$,^[1] so that we have

${\frac {d^{2}x_{\mu }}{ds}}=-\Gamma _{\alpha \beta }^{\mu }{\frac {dx_{\alpha }}{ds}}dx_{\beta }.$

(90)

We get the same line if we find the line which gives a stationary value to the integral

$\int ds{\text{ or }}\int {\sqrt {g_{\mu \nu }dx_{\mu }dx_{\nu }}}$

between two points (geodetic line).

1. The direction vector at a neighbouring point of the curve results, by a parallel displacement along the line element $(dx_{\beta })$ from the direction vector of each point considered.