without the Equivalence Hypothesis. 99 



the more exactly, the tensor is as in (a). On approaching 

 the sun we have instead, sensibly, 



. 2a 1 2a cc-f 2ax v v 2 * n , 



where r 2 = ,i\ 2 4- .r 2 2 + x s 2 and a = M/c 2 , which is about 1*5 km. ; 

 and this departure from the previous tensor cannot^ of 

 course, be transformed away ; the change due to the sun 

 is an essential one, a change of the metric properties all 

 around that body. The eqs. of motion of a particle which in 

 absence oE the sun were given by the geodesic 8\ds — with 

 the tensor (a), expressing uniform motion, are now again 

 given by the geodesic B\ds = with the modified tensor (6), 

 however. It is this system of eqs. which yielded the 

 remarkable result concerning the motion of the perihelion as 

 a welcome accessory of the classical planetary motion. But 

 what mainly interests us here is that according to Einstein's 

 theory the tensor gy is changed not only within the sun but 

 in all the circumjacent region of the world, the supplementary 

 terms fading away with distance. And similarly in the 

 presence of two or more lumps of " matter/'' which includes 

 not only ordinary matter but also the electromagnetic field, 

 for instance. The tensor components thus modified, as (b) 

 for instance, are (approximate) solutions of Einstein's " field 

 equations," certain generally covariant differential equations 

 of the seqond order written down by him in terms of a 

 tensor derived by contraction from the famous Riemann- 

 Ohristoffel tensor of rank four. The particular form of his 

 eqs. is here of no avail. Jt is enough to notice that, accord- 

 ing to these eqs , within matter not only the several gy but 

 also a certain differential invariant, the world-curvature, is 

 changed in value, while outside of matter the modified gij 

 are so distributed that the world-curvature remains nil as in 

 absence of matter. To repeat it, however, even outside of 

 matter the modification of gij is an essential one and cannot 

 be transformed away. 



To illustrate it by a bidimensional picture, imagine an ordinary 

 surface populated by one-dimensional beings using- one coordinate u for 

 their space or extension, and another v for their time ; their sun will be 

 a line segment, Aw, and the world-tube of the sun a certain strip of the 

 surface. Let our surface (and their "world"), in strict analogy to the 

 above, be an ordinary plane in absence of the sun ; then, in presence of 



* In Einstein's formulas (70), loc. cit. p. 819, u, is a misprint for 2 u, 

 as the reader will readily convince himself. 



H2 



