112 Dr. L. Silberstein on General Relativity 



itself of an ordinary two-dimensional sphere ; the differ- 

 ence, even with R 2 >0, being that the coordinate x± = l = it 

 is imaginary. Keeping this well in mind one can charac- 

 terize the required transformations by saying that any one 

 of them is a rotation of the world, sphere or pseudosphere, in 

 itself, similarly as the Lorentz transformations were described 

 as rotations of the Minkowskian, homaloidal world. Thus, 

 notwithstanding the world-curvature, the said group of 

 transformations is characterized in much the same way as 

 the Lorentz group (with one difference to be explained 

 presently). The details of its analytical expression will, of 

 course, be different for non-vanishing curvature. 



The result can now be stated shortly by saying that all the 

 natural sy stems of reference are derivable from one another 

 by a rotation of the world in itself, whatever its curvature. 

 To pass from a natural system of coordinates # ls etc., x i = I 

 to any non-natural, i* lv ..w 4 , is to distort the world sphere or 

 pseudosphere (without changing, however, its invariant 

 curvature), while to pass from that system to any other 

 natural system is to effect a mere rotation of the sphere or 

 pseudosphere, according to the sign of R 2 . The correspond- 

 ing group of transformations, deriving one natural system 

 from another, could appropriately be called the natural group, 

 of which then the Lorentz group would be a particular case 

 corresponding to i? 2 = co . It must be expressly stated that 

 I do not propose to limit the theory to the natural group : 

 on the contrary, I require every physical law to be covariant 

 (or contravariant) with respect to any transformations of the 

 coordinates. The " natural " ones are treated here at some 

 length only because of their eminently simple properties, as 

 a class of reference systems among an infinity of others. 



Now, as to the difference in relation to the Lorentz group, 

 hinted at a moment ago. It is well known that the so-called 

 general Lorentz transformations, viz. including pure space 

 rotations, constitute a s^-parametric group *, while our 

 natural group is a ^n-parametric one, since (21) are but 

 fifteen equations for the twenty-five coefficients a n , a 12 ,.-«55. 

 This, however, is only an apparent discrepancy. For the 

 Lorentz group just mentioned relates to a fixed origin of 

 x, y } z, I, or to put it shortly, to a fixed world-origin 0. 

 When we add the four degrees of freedom to choose a world- 

 point as 0, the result will precisely be 6 + 4 = 10. That is 



* The narrower or three-parametric Lorentz transformations do not 

 constitute a group, although they contain the subgroups for parallel 

 velocities. Of. the author's | Theory of Relativity/ Macmillan (1914), 

 p. 170. 



