98 Dr. L. Silberstein on General Relativity 



The gij thus constitute what is called a (co-variant) tensor of 

 rank two, Adz. a symmetrical one, since gij=gji- Any array 

 or matrix of 4x4 constituents aij which are transformed 

 according to the same rule is a covariant tensor of the 

 second rank. The differentials of the four coordinates them- 

 selves, which are transformed into 



3^5 



Lvttf/y ' -«y LvvLn* 



constitute a contravariant tensor of rank one or a four-vector. 

 The reader is supposed to be acquainted with these and 

 higher tensors and with their transformational properties *. 

 Here, therefore, it will be enough to recall that the import- 

 ance of all tensors consists in the linearity and homogeneity 

 of their transformation formulae ; whence, if all the con- 

 stituents of a tensor vanish in one system, they will vanish 

 also in an) r other system of coordinates (provided, of course, 

 that ~dx K fdx,! are not infinite). Thus, if a physical law is 

 written entirely in tensors, it will retain its form in passing 

 from one system of reference to any other. Tensors, and 

 tensors only, thus furnish the material for writing down such 

 laws. (This does not imply, of course, that they necessarily 

 will, but only that they may be obeyed by Nature.) The 

 fundamental tensor, gij, will manifestly play a prominent 

 part. 



Now, to come to our subject. In Einstein's theory the 

 tensor gij is intimately connected with gravitation so that 

 the latter codetermines the metrical properties of the world 

 or space-time. If there is no gravitation, or as we will say, 

 far away from heavy masses and disregarding the feeble con- 

 tribution due to electromagnetic and other energy, Einstein's 

 world, at least that of 1916, is Euclidean or homaloidal, 

 amounting to ds 2 = —cLv 1 2 — da' 2 2 — d l iY + d,r^^ x 4 =ct, or to 



#ii=#22=# , 33=--1j ^ = 1 (others zero). . . (a) 



In presence of gravitation this is changed. To make things 

 plain by an illustration, suppose there is but one conspicuous 

 body in the universe, say, our sun of mass M, the gravi- 

 tational contribution of a testing particle or "planet" being 

 negligible. Then, far away from the sun, and the farther 



* Those readers who are not familiar with the subject can inform 

 themselves in the easiest way by reading §§ 5-13 of Einstein's paper, 

 Ann. d. Physik, xlix. (1916), and Chaps. I. and II. of Wright's 

 'Invariants/ Cambr. Tract No. 9 0908). 



