Page:The principle of relativity (1920).djvu/235

 then we have

1 = g_{4 4}dx_{4}^2

dx_{4} = 1/[sqrt](g_{4 4}) = 1/[sqrt](1 + (g_{4 4} - 1)) = 1 - (g_{4 4} - 1)/2

or dx_{4} = 1 + k/8π [integral] ρdτ/r.

Therefore the clock goes slowly what it is placed in the neighbourhood of ponderable masses. It follows from this that the spectral lines in the light coming to us from the surfaces of big stars should appear shifted towards the red end of the spectrum.

Let us further investigate the path of light-rays in a statical gravitational field. According to the special relativity theory, the velocity of light is given by the equation

-dx_{1}^2 - dx_{2}^2 - dx_{3}^2 + dx_{4}^2 = 0;

thus also according to the generalised relativity theory it is given by the equation

(73) ds^2 = g_{μν} dx_{μ} dx_{ν} = 0.

If the direction, i.e., the ratio dx_{1}: dx_{2}: dx_{3} is given, the equation (73) gives the magnitudes

dx_{1}/dx_{4}, dx_{2}/dx_{4}, dx_{3}/dx_{4},

and with it the velocity,

[sqrt]((dx_{1}/dx_{4})^2 + (dx_{2}/dx_{4})^2 + (dx_{3}/dx_{4})^2) = γ,