﻿Einstein and Grossmann's Theory of Gravitation. 81 



lower. To cover the distance h a certain time is required, 



say - as a first approximation, when c denotes the velocity 



of light. Then the velocity v of the upper atom at the 

 moment of sending the signal is less than the velocity 



v -hy~ of the lower atom, when the signal arrives there. 



Thus the wavelets on arrival, according to Doppler's principle, 



will appear to have a wave-length which is / 1+ ~J times 



shorter. According to the equivalence hypothesis, the same 

 would be observed if the sodium atoms were at rest in a 

 svstem where a homogeneous gravitation field existed having 



a gradient ~- =7. This means that a vibrating sodium 



atom, compared with another that is placed in a spot where 

 the potential is less by an amount Ac£, would seem to vibrate 



( 1 H £■ \ times as fast as the latter. Thus, an observed 



terrestrial sodium line ought to be shifted to the violet, when 

 compared with a solar sodium line. 



Again, let a beam of light be sent at a certain moment by 

 a horizontal collimator at a point # = a, y = 0, z = b of the 

 moving system towards the axis of Z. To cover the distance 

 a, a certain time is required. When the beam of light is 

 observed by a telescope fixed to the Z-axis, this telescope will 



in the meanwhile have acquired an upward velocity v -\ — -, 



c 



if v is the velocity of the collimator at the moment of send- 

 ing the beam. According to the common aberration theory, 

 the telescope will have to be directed a little upward to 

 observe the beam. In the accelerated system the light-rays 

 will apparently be curved lines. By hypothesis, the same is 

 to be the case in a homogeneous gravitation field. In order 

 to account for this apparent rotation of the wave fronts we 

 must, following Huyghens's principle, conclude that the 

 velocity of light is greater in upper regions, where the 

 gravitation potential is higher. 



Of course these considerations are only approximate, but 

 they bring out important conclusions of the theory. 



The transformation from a resting into an accelerated 

 system of coordinates. 

 6. The equations which relate the coordinates and the 

 time, £, rj, f, and t, of the resting system to the coordinates 

 Phil Mag. S. G. Vol. 2D. No. L69. Jan. 1915. (J 



