Page:The principle of relativity (1920).djvu/234

 If the unit measuring rod lies on the x axis, the first of the equations (70) gives

g_{1 1} = -(1 + α/r).

From both these relations it follows as a first approximation that

(71) dx = 1 - α/2r.

The unit measuring rod appears, when referred to the co-ordinate-system, shortened by the calculated magnitude through the presence of the gravitational field, when we place it radially in the field.

Similarly we can get its co-ordinate-length in a tangential position, if we put for example

ds^2 = -1, dx_1 = dx_3 = dx_4 = 0, x_1 = r, x_2 = x_3 = 0

we then get

(71a) -1 = g_{2 2} dx_{2}^2 = -dx_{2}^2.

The gravitational field has no influence upon the length of the rod, when we put it tangentially in the field.

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, dx_1 = dx_2 dx_3 = 0;