258 



SCIENCE 



[N. S. Vol. LI. No. 1315 



Now, consider world-lines in our four dimen- 

 sional space. The complete record of all our 

 knowledge is a series of sequences of intersec- 

 tions of isuch. lines. By analogy I can draw in 

 ordinary space a great mimiber of intersecting 

 lines on a slieet of rubber ; I can tben bend and 

 deform the sheet to please myself ; by so doing 

 I do not introduce any new intersections nor 

 do I alter in the least the sequence of intersec- 

 tions. So in the space of our world-lines, the 

 space may be deformed in any imaginable man- 

 ner without introducing any new intersections 

 or changing the sequence of the existing inter- 

 sections. It is this sequence which gives us the 

 mathematical expression of our so-called ex- 

 perimental laws ; a deformation of our space is 

 equivalent mathematically to a transformation 

 of axes, consequently we see why it is that the 

 form of our laws must be the same when re- 

 ferred to any and all sets of axes, that is, must 

 remain unaltered by any mathematical trans- 

 formation. 



Now, ait last we come to gravitation. We can 

 not imagine any world-line simpler than that of 

 a particle of matter left to itself; we shall 

 therefore call it a " straight " line. Our experi- 

 ence is that two particles of matter attract one 

 another. Expressed in terms of world-lines, 

 this means that, if the world-lines of two iso- 

 lated particles come near each other, the lines, 

 instead of being straight, will be deflected or 

 bent in towards each other. The world-line of 

 any one iparticle is therefore deformed; and we 

 have just seen that a deformation is the equiva- 

 lent of a mathematical transformation. In 

 other words, for any one particle it is possible 

 to replace the effect of a gravitational field at 

 any instant by a mathematical transformation 

 of axes. The statement that this is always pos- 

 sible for any particle at any instant is Ein- 

 stein's famous " Principle of Equivalence." 



Let us rest for a moment, while I call atten- 

 tion to a most interesting coincidence, not to 

 be thought of as an intersection of world-lines. 

 It is siaid that Newton's thoughts were directed 

 to the observation of gravitational phenomena 

 by an apple falling on his head; from this 

 striking event he passed by natural steps to a 

 consideration of the universality of gravita- 



tion. Einstein in describing his mental proc- 

 ess in the evolution of his Mw of gravitation 

 says that his attention was called to a new 

 point of view by discussing his experiences 

 with a man whose fall from a high building he 

 had just witnessed. The man fortunately suf- 

 fered no serious injuries and assured Einstein 

 that in the course of his fall he had not been 

 conscious in the leiast of any pull downward on 

 his body. In mathematical language, with 

 reference to axes moving with the man the 

 force of gravity had disappeared. This is a 

 case where by the transfer of the axes from the 

 earth itself to the man, the force of the gravi- 

 tational field is annulled. The converse change 

 of axes from the falling man to a point on the 

 earth could be considered as introducing the 

 force of gravity into the equations of motion. 

 Another illustration of the introduction into 

 our equations of a force by means of a change 

 of axes is furnished by the ordinary treatment 

 of a body in uniform rotation about an axis. 

 For instance, in the case of a so-called conical 

 pendulum, that is, the motion of a bob sus- 

 pended from a fixed point by a string, which is 

 so set in motion that the bob describes a hori- 

 zontal circle and the string therefore describes 

 a circular cone, if we transfer our axes from 

 the earth and have them rotate around the ver- 

 tical line through the fixed point with the 

 same angular velocity as the bob, it is neces- 

 sary to introduce into our equations of motion 

 a fictitious " force " called the centrifugal 

 force. No one ever thinks of this force other 

 than as a mathematical quantity introduced 

 into the equations for the siake of simplicity of 

 treatment; no physical meaning is attached to 

 it. Why should there be to any other so-called 

 " force," which, like centrifugal force, is inde- 

 pendent of the nature of the matter? Again, 

 here on the earth our sensation of weight is 

 interpreted mathematically by combining ex- 

 pressions for centrifugal force and gravity; we 

 have no distinct sensation for either separately. 

 Why then is there any difference in the essence 

 of the two? Why not consider them both as 

 brought into our equations by the agency of 

 mathematical transformations? This is Ein- 

 stein's point of view. 



