230 Professor A. S. Eddington [Feb. 1, 



Bent a centrifugal force. Clearly the hypothesis of equivalence requires 

 that there should be one or more general differential equations satisfied 

 by the ^'s in all cases, and not a special law satisfied by gravitational 

 ^'s and another satisfied by distortion ^'s. If, then, the law is a general 

 relation between the g's, it must hold for all systems of co-ordinates ; 

 that is, it must be co-variant for all transformations. 



The general condition satisfied by the ^'s in the alseiice of a gravita- 

 tional field is written in the form — 



B^" = (5) 



IXCTT 



where the quantity on the left is known as the Kiemann-Christoffel 

 tensor. (The word tensor expresses the property, that if it vanishes in 

 one system of co-ordinates it vanishes in all systems.) It is a function of 

 the ^'s and their first and second derivatives with respect to x^, x.^, x., x^. 

 It has 256 components, formed by ringing the changes on the suffixes 

 p, fx, 0-, r, giving them the values 1, 2, 3, 4.* By s^nnmetry many of 

 these are identical, and we actually get (I think) 96 apparently different 

 equations, some of which may not be independent. The equation (5) 

 is to be understood to mean that all the 96 components vanish. The 

 equation expresses the fact that a mathematical transformation exists 

 which can transform (3) to (2) throughout space ; and that is the method 

 by which it is obtained analytically. 



The general equation between the ^'s, allowing for a gravitational field, 

 must be less stringent ; it must be such that it is satisfied when (5) is 

 satisfied, but not necessarily vice versa. (Zero gravitation is a particular 

 case of gravitation, but not vice versa.) The simplest symmetrical law 

 that we could propose is — 



Bao- = 2 B^" =0 (6) 



p.l /^^^ 



This is clearly satisfied when (5) is satisfied. 



B/xo- is called the reduced (verjiinf/t) Eiemann-Christoffel tensor, and 

 has ten different (but not all independent) components. It seems to be 

 the only possible way of symmetrically building up another tensor out 



of the components of B^ ; and it appears also that equation (6) is the 



only CO- variant equation of the second rank (i.e., having ten components) 

 that can be formed from the ^'s, and their first and second derivatives 

 and linear in the last. Co-variant equations of higher rank (with more 

 components) would impose too great restrictions, and like the Riemann- 

 Christoffel tensor would not admit a gravitational field. 



For this reason (6) is chosen as the new law of gravitation. It 

 reduces to the Newtonian law as first approximation. 



It remains to see how the equation (6) is modified when the space 

 is occupied by mass, i.e. electromagnetic energy. What is to be 



* It would be cumbrous to write down the value of B^ ; but it will be 



fXffT 



understood that it contains g , x , etc., and the different components are 

 got by giving the values 1, 2, 3, 4 to the suffiiLes. 



