Fundamental Law of Electrical Action. 

 6. The Ponder omotive Force *. 



357 



If a be the potential-four- vector in an electric field, and p 

 be the electric space-density at a point, the force acting on 

 this point is given by the matrix 



uu 



u 2 , 



"3, 



2 





3y' 



3- ' 



92 



a l? 



a 2) 



a 35 



a 4 



(7) 



It should be noticed that the word " Force " is used in a 

 generalized sense. The components of this four-vector are 

 (X, Y, Z) the ordinary space-components, and 



L = i(Kiii + Yu 2 + Zu 3 ), 



i. £., v / — l times the rate of doing work. The four com- 

 ponents are connected by the equation 



Xzt?! + Yw 2 + Zw 3 + Lu> 4 = 0, 



i. e., the force-four- vector is always normal to the velocity- 

 four-vector. 



Writing (<£, F, G, H) for W — la 4 , a 1? a 2 , a 3 ) and intro- 

 ducing the ordinary C.Gr.S. units, it can be easily verified 

 that this expression is identical with Lorentz's expressions 

 for Ponderomotive Force. 



We shall now write (w^ w 2 , w z , w^) instead of (u 1? u 2 , 

 u 3 , V— !)• Then 



[X,Y,Z,L]= Po 



w l 



w 2 



w s 



w± 





3y 



B 



B* 



B 



3* 



aj 



ag 



a 3 



a 4 



p =p v /(l — u 2 ) is an invariant, and is generally known as 

 the rest-density, 



po'w' 



and 



(a 1? a 2 , a 3 , a 4 ) 



R' 



R' = perpendicular distance from the external point 

 (x y y, z, I) on the axis of motion of the charge p\ which 

 produces the field. 



* Minkowski, he. cit. § 11. 



