i£i^OJ The Theory of Relativity 31 



suppose the wire rope breaks ; the elevator v^^ith its contents falls ver- 

 tically with the gravitational acceleration of thirty-two feet per sec- 

 ond per second. 



If we could keep our mental equilibrium under the circum- 

 stances, we would notice that our weight appeared to vanish ; that the 

 package we carried also lost its weight, and if we removed our hand 

 from it, it would remain suspended in mid-air ; the pendulum, if at 

 the end of its swing when the rope broke, would remain there motion- 

 less and would not swing back. In other words, so far as we are 

 concerned, and so long as the elevator is falling, gravity has been anni- 

 hilated by giving the appropriate acceleration to the elevator. In 

 reality, an observer inside a closed windowless box could not by any 

 means decide whether the box is in a static gravitational field, or is 

 endowed with accelerated motion in a space free from gravitation. 

 We seem to weigh more while in an elevator ascending with acceler- 

 ated motion, so that acceleration simulates gravitation and may be 

 substituted for it. Now imagine two sets of four-dimensional axes, one 

 set stationarj^ in the Woolworth building, and the other set fixed in 

 the falling elevator. Let x, y, z, and t be the co-ordinates of a point 

 with reference to the second, or falling-elevator set of axes. Then, 

 since we have no gravitation in the elevator to complicate matters, an 

 element of length ds with reference to this set would be given by 

 the equation 



fls- =: dx- + cly' + dz= + dt- 



Now let the same point mentioned above have at the same instant 

 the co-ordinates x', y', z' and t' with reference to the first, or fixed-in- 

 the-building set of axes. Then the transformation equation for ds 

 will be 



d8^ — g„dx'2 + g^dj'' + g,,dz" + g44clt'- + 2g,A x'dy' -f 2g,,dx 'dz' + etc. 



There are ten of these g coefficients, and their value depend? on 

 ihe nature of the transformation, and specifies it. They can be used, 

 therefore, not only to specify it, but they also define the original grav- 

 itational field, because they specify how it was got rid of. 



Now in Einstein's theory these ten gr's, used in ten differential 

 equations, are regarded as ten values of the gravitational potential 

 specifying the field, and one of them, g^^, is approximately the same 

 as the Newtonian potential, except for a factor. 



