February 17, 192 1] 



NATURE 



791 



freely moved about so as to remain in contact with 

 the surface. 



To avoid misunderstanding, it should be said 

 that Riemann's expression for ds' is not the only 

 one that is taken to be the typical or standard 

 formula. The important thing- is that, given any 

 formula for ds^, in a P„, we can, by direct calcula- 

 tion, find an expression for the curvature of Pn in 

 the neighbourhood of any assigned point (x). It 

 is only when this curvature is everywhere the 

 same that we have a P« for which the axiom of 

 free mobility is valid. When the curvature varies 

 from place to place we are not entitled, for in- 

 stance, to assume that we can carry about an 

 invariable foot-rule for purposes of physical 

 measurement. 



In the simpler theory of relativity we have a 

 formula 



di2 = (f.t:'!-l-dy'-|-dz2-c«dt', ... (i) 



where c is a real constant. As it originally pre- 

 sents itself, X, y, z are ordinary rectangular co- 

 ordinates, t is the time, and c the experimental 

 velocity of light. By a suitable choice of units we 

 can make the value of c any finite constant that 

 we please. Following Minkowski, I shall call 

 (x, y, s. t) a world-point; the aggregate of these 

 points may be provisionally called a space-time 

 world P(.x, y, s, t). 



When t = tQ, a constant, dt = o and (i) reduces 

 to the ordinary Euclidean formula. We may ex- 

 press this by saying that the sub-world 

 P{x. y, z, to) is Euclidean. Actual experiments 

 take time ; so we cannot verify this assertion by 

 observation. If, however, two observers, at dif- 

 ferent places, make measurements which begin 

 and end at the same instants, we may expect their 

 results to be consistent. As Prof. Einstein has 

 pointed out, the question of simultaneity (and, 

 indeed, of time itself, as an observed quantity) is 

 a more diflTicult one than appears at first sight. 



The main difficulty about (i), as it seems to me, 

 is that the expression on the right is not a definite 

 form ; hence in the neighbourhood of every "real " 

 point (x, y, z, i) there is a real region for which 

 ds* is negative. It is possible that the difficulty 



of interpretation is more apparent than real, as is 

 the case in some well-known examples. For in- 

 stance, a hyperbola may be analytically defined as 

 an ellipse of semiaxes a, bi, where a, b are real; 

 and, moreover, v. Staudt's theory of involution 

 gives an actual geometrical meaning to the alge- 

 braic definition. 



If, with »'*= — I, we put ct = iT, the formula (i) 

 becomes 



ds3 = d.x2-(-dy2 + d2« + dr», ... (2) 



the typical formula for a Euclidean P4. This makes 

 it very tempting to assume that the successions 

 of phenomena in our world of experience are, so 

 to speak, sections of a space-world P{x, y, s, r), 

 obtained by giving r purely imaginary values. 

 This point of view has been taken by Minkowski 

 and others. 



The mathematical theories of abstract geometry 

 and kinematics are so complete that physicists 

 have a definite set of hypotheses from which to 

 choose the one most suited to their purpose; and 

 besides this they have to frame axioms and defini- 

 tions about time, energy, etc., with which the pure 

 mathematician is not concerned. 



Whatever may be the ultimate form given to 

 the theory of relativity, the predictive quality of 

 its formulae gives it a high claim to attention, and 

 it certainly seems probable that, for the sake of 

 what Mach calls economy of thought, we may feel 

 compelled to change our ideas of " actual " space 

 and time. 



In an article like this it is impossible to go into 

 detail ; the following references may be useful to 

 readers who desire further information : — " The Ele- 

 ments of Non-Euclidean Geometry," by J. L. 

 Coolidge, is rather condensed, but very conscientious 

 and trustworthy ; one of the best analytical discus- 

 sions of the metrical theory is in Bianchi's " Lczioni 

 di Geometria Diflferenziale," chap, xi.; and Lie's 

 "Theorie der Transformationsgruppen," vol. iit., 

 chaps, xx.-xxiv., contains a most valuable critique of 

 Riemann and Hclmholtz. The article "Geometry" in 

 the " Encyclopaedia Britannicn " (last edition) gives an 

 outline of the theory and numerous references. 

 Finallv, there is an elaborate " Bibliography of Non- 

 Euclidean Geometry " by D. M. J. Some'rville (se« 

 Naturk, May 16, 1912, vol. Ixxxix., p. 266). 



The General Physical Theory of RelatiTity. 

 By J. H. Jeans, Sec. R.S. 



THE relativity theory of gravitation, which is 

 at present the centre of so much interest, 

 owes its existence to an earlier physical theory of 

 relativity which had proved to be in accord with 

 all the known phenomena of Nature except gravi- 

 tation. The gravitational theory is only one 

 branch, although a vigorous and striking branch, 

 of a firmly established parent tree. The present 

 article will deal solely with the main trunk and 

 roots of this tree. 



Newton's laws of motion referred explicitly to 

 a state of rest, but also showed that the pheno- 

 mena to be expected from bodies in a state of rest 



NO. 2677, VOL. 106] 



were precisely identical with those to be expected 



j when the same bodies were moving with constant 



I velocity. Indeed, Newton directed special atten- 



; tion to this implication of his laws of motion in the 



following words : — 



Corollary V. : The motions of bodies included 

 in a given space are the same among themselvts, 

 whether that space is at rest, or moves uniformly 

 forwards in a right line without any circular 

 motion. 



"A clear proof of which we have," continues 

 Newton, "from the experiment of a ship, where 

 all motions happen after the same manner whether 



n 



