448 



NATURE 



.->i.i i \. 



22, 1923 



endoparasitic organisms, will determine largely the 

 extent to which he can use and develop the natural 

 resources of the rich tropical and sub-tropical zone of 

 the earth. 



Other applications of /oology to human well-l>eing 

 cannot be dealt with here, but mention should be made 

 of two— the researches on sea-fisheries problems which 



have formed an important branch of the zoological 

 work of Great Britam for forty years, and the studies 

 on genetics which made possible an expl ' tIk 



mode of inheritance of a particular bli aii<i 



of some of the defects {e.g. colour! .:■ md 

 haemophilia) and malformations which \x\^\ 

 human race. 



in U.< 



The Theory of the Affine Field.' 

 By Prof. Albert Einstein, For. Mem. R.S. 



''F'HE theory' of the connexion between gravitation 

 J- and clc( tromagnetism outlined below is founded 

 on Eddington's idea, published during recent years, of 

 basing " field physics " mathematically on the theory 

 of the afline relation. We shall first briefly consider 

 the entire development of ideas associated with the 

 names Levi-Civita, VVeyl, and Eddington. 



The general theory of relativity rests formally on 

 the geometry of Riemann, which bases all its concep- 

 tions on that of the interval ds between points in- 

 definitely near together, in accordance with the 

 formula^ rf.»=^,^x,^x (i) 



These magnitudes ^^^ determine the behaviour of 

 measuring-rods and clocks with reference to the co- 

 ordinate system, as well as the gravitational field. 

 Thus far we are able to say that, from its foundations, 

 the general theory of relativity explains the gravita- 

 tional field. In contrast to this, the conceptual founda- 

 tions of the theory have no relations with the electro- 

 magnetic field. 



These facts suggest the following question. Is it not 

 possible to generalise the mathematical foundations of 

 the theory in such a way that we can derive from them 

 not only the properties of the gravitational field, but 

 also those of the electromagnetic field ? 



The possibility of a generalisation of the mathe- 

 matical foundations resulted from the fact that Levi- 

 Civita pointed out an element in the geometry of 

 Riemann that could be made independent of this 

 geometry, to wit, the " afline relation " ; for according 

 to Riemann's geometry every indefinitely small part of 

 the manifold can be represented approximately by a 

 Euclidean one. Thus in this elemental region there 

 exists the idea of parallelism. If we subject a con- 

 travariant vector A'' at the point x^ to a parallel 

 displacement to the indefinitely adjacent point x„-i-Sx„, 

 then the resulting vector ^''-h8.(4<^ is determined by an 

 expression of the form 



M'^=-VlA^hxy 



(2) 



The magnitudes F are symmetrical in the lower indices, 

 and are expressed in accordance with Riemann 

 geometry by the g^^ and their first derivatives 

 (Christoffel symbols of the second kind). We obtain 

 these expressions by formulating the condition that 

 the length of a contravariant vector formed in 

 accordance with (i) does not change as a result of the 

 parallel displacement. 



Levi-Civita has shown that the Riemann tensor of 

 curvature, which is fundamental for the theory- of the 



» Translated by Dr. R. W. Lawson. 



* In accordance with custom, the signs of summation are omitted. 



gravitational field, can be obtained from a geometrical 

 consideration based .solely on the law of the affine 

 relation given by (2) above. The manner in which the 

 r^"^ are expressible in terms of the ^^^ plays no part 

 in this consideration. The behaviour in the case of 

 differential operations of the absolute differential 

 calculus is analogous. 



These results naturally lead to a generalisation of 

 Riemann's geometry. Instead of starting off from the 

 metrical relation (i) and deriving from this the co- 

 efficients r of the aflSne relation characterised by (2), 

 we proceed from a general affme relation of the t>'pe 

 (2) without postulating (i). The search for the 

 mathematical laws which shall correspond to the laws 

 of Nature then resolves itself into the solution of the 

 question : What are the formally most natural con- 

 ditions that can be imposed upon an affine relation ? 



The first step in this direction was taken by H. Weyl. 

 His theor\' is connected with the fact that light rays 

 are simpler structures from the physical view-point 

 than measuring-rods and clocks, and that only the 

 ratios of the g^^, are determined by the law of pro- 

 pagation of light. Accordingly he ascribes objective 

 significance not to the magnitude ds in (i), i.e. to the 

 length of a vector, but only to the ratio of the lengths 

 of two vectors (thus also to the angles). Those affine 

 relations are permissible in which the parallel displace- 

 ment is angularly accurate. In this way a theory' was 

 arrived at, in which, along with the determinate (except 

 for a factor) g^y other four magnitudes <^,. occurred, 

 which Weyl identified with electromagnetic potentials. 

 Eddington attacked the problem in a more radical 

 manner. He proceeded from an affine relation of the 

 type (2) and sought to characterise this without intro- 

 ducing into the basis of the theory anything derived 

 from (i), i.e. from the metric. The metric was to 

 appear as a deduction from the theor>'. The tensor 



Jl'po gp a 



r" 



^ aH 



(3) 



NO. 2812, VOL. I 12] 



is symmetrical in the special case of Riemann s 

 geometry. In the general case 7?^^ is split up into a 

 s}Tnmetrical and an " anti-symmetrical " part : 



^M.' = 7>*.' + ^M»' • (4) 



One is confronted with the possibility of identifying 

 7^y with the symmetrical tensor of the metrical or 

 gravitational field, and ^^„ with the antis\-mmetrical 

 tensor of the electromagnetic field. This was the 

 course taken by Eddington. But his theor)' remained 

 incomplete, because at first no course jx)ssessed of the 

 advantages of simplicity and naturalness presented 



