THE LAW OF MOTION 125 



defined on a curved surface without reference to any 

 frame or system of partitions, viz. the geodesies or 

 shortest routes from one point to another. The geo- 

 desies of our curved space-time supply the natural tracks 

 which particles pursue if they are undisturbed. 



We observe a planet wandering round the sun in an 

 elliptic orbit. A little consideration will show that if we 

 add a fourth dimension (time), the continual moving on 

 in the time-dimension draws out the ellipse into a helix. 

 Why does the planet take this spiral track instead of 

 going straight? It is because it is following the shortest 

 track; and in the distorted geometry of the curved 

 region round the sun the spiral track is shorter than any 

 other between the same points. You see the great 

 change in our view. The Newtonian scheme says that 

 the planet tends to move in a straight line, but the sun's 

 gravity pulls it away. Einstein says that the planet tends 

 to take the shortest route and does take it. 



That is the general idea, but for the sake of accuracy 

 I must make one rather trivial correction. The planet 

 takes the longest route. 



You may remember that points along the track of 

 any material body (necessarily moving with a speed less 

 than the velocity of light) are in the absolute past or 

 future of one another; they are not absolutely ''else- 

 where". Hence the length of the track in four dimensions 

 is made up of time-like relations and must be measured 

 in time-units. It is in fact the number of seconds 

 recorded by a clock carried on a body which describes 

 the track.* This may be different from the time re- 



* It may be objected that you cannot make a clock follow an arbitrary 

 curved path without disturbing it by impressed forces (e.g. molecular 

 hammering). But this difficulty is precisely analogous to the difficulty 

 of measuring the length of a curve with a rectilinear scale, and is sur- 

 mounted in the same way. The usual theory of "rectification of curves" 

 applies to these time-tracks as well as to space-curves. 



