Page:The Meaning of Relativity - Albert Einstein (1922).djvu/91

Rh Owing to the symmetry of the expression in the brackets with respect to the indices $$\mu$$ and $$\nu$$, this equation can be valid for an arbitrary choice of the vectors $$(A^\mu)$$ and $$dx_\nu$$ only when the expression in the brackets vanishes for all combinations of the indices. By a cyclic interchange of the indices $$\mu, \nu, \alpha$$, we obtain thus altogether three equations, from which we obtain, on taking into account the symmetrical property of the $$\Gamma_{\mu\nu}^\alpha$$,

in which, following Christoffel, the abbreviation has been used,

If we multiply (68) by $$g^{\alpha\sigma}$$ and sum over the $$\alpha$$, we obtain

in which $$\begin{Bmatrix}\mu\nu \\ \sigma\end{Bmatrix}$$ is the Christoffel symbol of the second kind. Thus the quantities $$\Gamma$$ are deduced from the $$g_{\mu\nu}$$. Equations (67) and (70) are the foundation for the following discussion.

Co-variant Differentiation of Tensors. If $$(A^\mu + \delta A^\mu)$$ is the vector resulting from an infinitesimal parallel displacement from $$P_1$$ to $$P_2$$, and $$(A^\mu + dA^\mu$$ the vector $$A^\mu$$ at the point $$P_2$$ then the difference of these two,