Page:The Meaning of Relativity - Albert Einstein (1922).djvu/101

where the $\gamma _{\mu \nu }$ are to be regarded as small of the first order.

Both terms of our equation of motion are then small of the first order. If we neglect terms which, relatively to these, are small of the first order, we have to put

$ds^{2}=dx_{\nu }^{2}=dl^{2}(1-q^{2})$

(93)

$\Gamma _{\alpha \beta }^{\mu }=-\delta _{\mu \sigma }{\begin{bmatrix}\alpha \beta \\\sigma \end{bmatrix}}=-{\begin{bmatrix}\alpha \beta \\\mu \end{bmatrix}}={\frac {1}{2}}\left({\frac {\delta \gamma _{\alpha \beta }}{\delta x_{\mu }}}-{\frac {\delta \gamma _{\alpha \mu }}{\delta x_{\beta }}}-{\frac {\delta \gamma _{\beta \mu }}{\delta x_{\alpha }}}\right).$

(94)

We shall now introduce an approximation of a second kind. Let the velocity of the material particles be very small compared to that of light. Then $ds$ will be the same as the time differential, $dl$. Further, ${\frac {dx_{1}}{ds}},{\frac {dx_{2}}{ds}},{\frac {dx_{3}}{ds}}$ will vanish compared to \frac{dx_4}{ds}. We shall assume, in addition, that the gravitational field varies so little with the time that the derivatives of the $\gamma _{\mu \nu }$ by $x_{4}$ may be neglected. Then the equation of motion (for $\mu =1,2,3$) reduces to

${\frac {d^{2}x_{\mu }}{dl^{2}}}={\frac {\delta }{\delta x_{\mu }}}\left({\frac {\gamma _{44}}{2}}\right).$

(90a)

This equation is identical with Newton's equation of motion for a material particle in a gravitational field, if we identify $\left({\frac {\gamma _{44}}{2}}\right)$ with the potential of the gravitational field; whether or not this is allowable, naturally depends upon the field equations of gravitation, that is, it depends upon whether or not this quantity satisfies, to a first approximation, the same laws of the field as the