Page:The Meaning of Relativity - Albert Einstein (1922).djvu/90

78 character, that they must depend linearly and homogeneously upon the $$dx_\nu$$ and the $$A^\nu$$. We therefore put

In addition, we can state that the $$\Gamma_{\alpha\beta}^\nu$$ must be symmetrical with respect to the indices $$\alpha$$ and $$\beta$$. For we can assume from a representation by the aid of a Euclidean system of local co-ordinates that the same parallelogram will be described by the displacement of an element $$d^{(1)}x_\nu$$ along a second element $$d^{(2)}x_\nu$$ as by a displacement of $$d^{(2)}x_\nu$$ along $$d^{(1)}x_\nu$$. We must therefore have

The statement made above follows from this, after interchanging the indices of summation, $$\alpha$$ and $$\beta$$, on the right-hand side.

Since the quantities $$g_{\mu\nu}$$ determine all the metrical properties of the continuum, they must also determine the $$\Gamma_{\alpha\beta}^\nu$$. If we consider the invariant of the vector $$A^\nu$$, that is, the square of its magnitude,

which is an invariant, this cannot change in a parallel displacement. We therefore have

or, by (67),