Page:The Meaning of Relativity - Albert Einstein (1922).djvu/100

88 In fact, this equation reduces to that of a straight line if all the components, $$\Gamma_{\alpha\beta}^\mu$$, of the gravitational field vanish.

How are these equations connected with Newton's equations of motion? According to the special theory of relativity, the $$g_{\mu\nu}$$ as well as the $$g^{\mu\nu}$$, have the values, with respect to an inertlal system (with real time co-ordinate and suitable choice of the sign of $$ds^2$$), {{MathForm2|(91)| $$ \left. \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right\}. $$}} The equations of motion then become

We shall call this the "first approximation" to the $$g_{\mu\nu}$$-field. In considering approximations it is often useful, as in the special theory of relativity, to use an imaginary $$x_4$$-co-ordinate, as then the $$g_{\mu\nu}$$, to the first approximation, assume the values {{MathForm2|(91a)| $$ \left. \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{matrix} \right\}. $$}} These values may be collected in the relation

To the second approximation we must then put