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

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 $ds^{2}$),

$\left.{\begin{matrix}-1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&1\end{matrix}}\right\}.$

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The equations of motion then become

${\frac {d^{2}x_{\mu }}{ds^{2}}}=0.$

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

$\left.{\begin{matrix}-1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{matrix}}\right\}.$

(91a)

These values may be collected in the relation

$g_{\mu \nu }=-\delta _{\mu \nu }.$

To the second approximation we must then put

$g_{\mu \nu }=-\delta _{\mu \nu }+\gamma _{\mu \nu }$

(92)