Page:The Meaning of Relativity - Albert Einstein (1922).djvu/97

refers to the point $P$. Subtracting ${\frac {1}{2}}d(\xi ^{\alpha }\xi ^{\beta })$ from the integrand, we obtain

${\frac {1}{2}}\int _{0}(\xi ^{\alpha }d\xi ^{\beta }-\xi ^{\beta }d\xi ^{\alpha }).$

This skew-symmetrical tensor of the second rank, $f_{\alpha \beta }$, characterizes the surface element bounded by the curve in magnitude and position. If the expression in the brackets in (85) were skew-symmetrical with respect to the indices $\alpha$ and $\beta$, we could conclude its tensor character from (85). We can accomplish this by interchanging the summation indices $\alpha$ and $\beta$ in (85) and adding the resulting equation to (85). We obtain

$2\Delta A^{\mu }=-R_{\sigma \alpha \beta }^{\mu }A^{\sigma }f^{\alpha \beta }$

(86)

in which

$R_{\sigma \alpha \beta }^{\mu }=-{\frac {\delta \Gamma _{\sigma \alpha }^{\mu }}{\delta x_{\beta }}}+{\frac {\delta \Gamma _{\sigma \beta }^{\mu }}{\delta x_{\alpha }}}+\Gamma _{\rho \alpha }^{\mu }\Gamma _{\sigma \beta }^{\rho }-\Gamma _{\rho \beta }^{\mu }\Gamma _{\sigma \alpha }^{\rho }.$

(87)

The tensor character of $R_{\sigma \alpha \beta }^{\mu }$ follows from (86); this is the Riemann curvature tensor of the fourth rank, whose properties of symmetry we do not need to go into. Its vanishing is a sufficient condition (disregarding the reality of the chosen co-ordinates) that the continuum is Euclidean.

By contraction of the Riemann tensor with respect to the indices $\mu$, $\beta$, we obtain the symmetrical tensor of the second rank,

$R_{\mu \nu }=-{\frac {\delta \Gamma _{\mu \nu }^{\alpha }}{\delta x_{\alpha }}}+\Gamma _{\mu \beta }^{\alpha }\Gamma _{\nu \alpha }^{\beta }+{\frac {\delta \Gamma _{\mu \alpha }^{\alpha }}{\delta x_{\nu }}}-\Gamma _{\mu \nu }^{\alpha }\Gamma _{\alpha \beta }^{\beta }.$

(88)

The last two terms vanish if the system of co-ordinates