(71) _dx_ = 1 - α/2_r_.
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
_ds²_ = -1, _dx₁_ = _dx₃_ = _dx₄_ = 0, _x₁_ = _r_, _x₂_ = _x₃_ = 0
we then get
(71a) -1 = _g₂₂_ _dx₂²_ = -_dx₂²_.
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, _dx₁_ = _dx₂_ _dx₃_ = 0;
then we have