Page:The principle of relativity (1920).djvu/232

 The last of the equations (53) thus leads to

(68) [nabla]^2 g_{4 4} = κρ.

The equations (67) and (68) together, are equivalent to Newton's law of gravitation.

For the gravitation-potential we get from (67) and (68) the exp.

(68a.) -κ/(8π) [integral]ρdτ/r

whereas the Newtonian theory for the chosen unit of time gives

-K/c^2 [integral]ρdτ/r, where K denotes usually the gravitation-constant. 6.7 x 10^{-8}; equating them we get

(69) κ = 8πK/c^2 = 1.87 x 10^{-27}.

§22. Behaviour of measuring rods and clocks in a statical gravitation-field. Curvature of light-rays. Perihelion-motion of the paths of the Planets.

In order to obtain Newton's theory as a first approximation we had to calculate only g_{4 4}, out of the 10 components g_{μν} of the gravitation-potential, for that is the only component which comes in the first approximate equations of motion of a material point in a gravitational field.

We see however, that the other components of g_{μν} should also differ from the values given in (4) as required by the condition g = -1.