Page:The Meaning of Relativity - Albert Einstein (1922).djvu/123

We shall now show that these three effects, which are to be expected in accordance with Mach's ideas, are actually present according to our theory, although their magnitude is so small that confirmation of them by laboratory experiments is not to be thought of. For this purpose we shall go back to the equations of motion of a material particle (90), and carry the approximations somewhat further than was done in equation (90a).

First, we consider $\gamma _{44}$ as small of the first order. The square of the velocity of masses moving under the influence of the gravitational force is of the same order, according to the energy equation. It is therefore logical to regard the velocities of the material particles we are considering, as well as the velocities of the masses which generate the field, as small, of the order ${\frac {1}{2}}$. We shall now carry out the approximation in the equations that arise from the field equations (101) and the equations of motion (90) so far as to consider terms, in the second member of (90), that are linear in those velocities. Further, we shall not put $ds$ and $dl$ equal to each other, but, corresponding to the higher approximation, we shall put

$ds={\sqrt {g_{44}}}dl=\left(1-{\frac {\gamma _{44}}{2}}\right)dl.$

From (90) we obtain, at first.

${\frac {d}{dl}}\left[\left(1+{\frac {\gamma _{44}}{2}}\right){\frac {dx_{\mu }}{dl}}\right]=-\Gamma _{\alpha \beta }^{\mu }{\frac {dx_{\alpha }}{dl}}{\frac {dx_{\beta }}{dl}}\left(1+{\frac {\gamma _{44}}{2}}\right).$

(116)

From (101) we get, to the approximation sought for.