Thus Euclidean geometry does not hold in the gravitational field even in the first approximation, if we conceive that one and the same rod independent of its position and its orientation can serve as the measure of the same extension. But a glance at (70a) and (69) shows that the expected difference is much too small to be noticeable in the measurement of earth’s surface.
We would further investigate the rate of going of a unit-clock which is placed in a statical gravitational field. Here we have for a period of the clock
ds = 1, dx₁ = dx₂ dx₃ = 0;
then we have
1 = g₄₄dx₄²
dx₄ = 1/√(g₄₄) = 1/√(1 + (g₄₄ - 1)) = 1 - (g₄₄ - 1)/2
or dx₄ = 1 + k/8π ∫ 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
-dx₁² - dx₂² - dx₃² + dx₄² = 0;