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

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

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

in which

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,

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