A:\-2 Woolard— Generalized Relativity and Gravitation. 



Therefore the differentia] of are is unchanged by such a 

 transformation; but by the equivalence hypothesis, such 

 a transformation is the same as would be produced by 

 the neat ion of an homogeneous gravitation field in the 

 system. It is reasonable to assume that a free particle 

 always takes the shortest possible track between two 

 points, and that (11) is invariant for any arbitrary 

 transformation corresponding to the creation of any 

 tmhomogeneous gravitation field. Executing the trans- 

 formation (5) we see that. the //,,■ determine the gravita- 

 tion field and all its characteristics. 



The determining feature of a gravitation field is its 

 strength, which is measured by the potential. The poten- 

 tial energy of a unit mass at any point is the grav- 

 itational potential at that point, and is equal to the 

 work done in bringing the mass from infinity to that 

 point ; hence the gravitational potential always has 

 a negative sign, since in being so moved, a body itself 

 does work. In Newtonian mechanics we have only one 

 potential, but in this new theory, there are, of course, ten 

 potentials, one of which, under ordinary conditions, is 

 closely approximate to the Newtonian potential.^ 



11. The ten coefficients may be written 



9u 9 t , 9,. 9 lt ] 



9 ;i 9 ; , 9 ; , ff„ g = g l (12) 



/7 3 , 9ti #33 #34 



9* 9* r/,3 ff« J 



This puts them into the form of a mathematical operator, 

 known as a tensor, by means of which our various trans- 

 formations and operations may be performed. It is 

 impossible in this paper to go into the complex mathe- 

 matics of these tensors ; the original papers may be con- 

 sulted by those who have the necessary mathematical 

 knowledge 13 ; suffice it to say, that if a transformation of 

 coordinates be executed upon a tensor, each component, 

 g j of the tensor transforms itself exactly as the corre- 

 sponding component of the differential of arc, dx.dxj- 

 Hence if we have any vector with components Ft, and any 

 tensor with components T ; , and if it be true that 



T, = F, (13) 



13 Einstein and Grossmann, Zeitsehr. fur Math. u. Physik, 1914, Jan. 

 Einstein, Sitzungber. Berlin, 1914, Nov.; ibid., 1915, Nov. 25. 



