Page:Eddington A. Space Time and Gravitation. 1920.djvu/188

statics. Surely it is more than a coincidence that the physicist needs just four more quantities to specify the state of the world at a point in space, and four more quantities are provided by removing a rather illogical restriction on our system of geometry of natural measures.

[The general reader will perhaps pardon a few words addressed especially to the mathematical physicist. Taking the ordinary unaccelerated rectangular coordinates $x$, $y$, $z$, $t$, let us write $F$, $G$, $H$, $-\Phi$ for $\kappa _{1}$, $\kappa _{2}$, $\kappa _{3}$, $\kappa _{4}$, then ${\frac {dl}{l}}=\lambda =F\,dx+G\,dy+H\,dz-\Phi \,dt$. From which, by integration, $\log l+{\text{const.}}=\int (F\,dx+G\,dy+H\,dz-\Phi \,dt)$.

The length $l$ will be independent of the route taken if $F\,dx+G\,dy+H\,dz-\Phi \,dt$ is a perfect differential. The condition for this is ${\begin{aligned}{\frac {\partial H}{\partial y}}-{\frac {\partial G}{\partial z}}&=0{\text{,}}&{\frac {\partial F}{\partial z}}-{\frac {\partial H}{\partial x}}&=0{\text{,}}&{\frac {\partial G}{\partial x}}-{\frac {\partial F}{\partial y}}&=0{\text{,}}\\-{\frac {\partial \Phi }{\partial x}}-{\frac {\partial F}{\partial t}}&=0{\text{,}}&-{\frac {\partial \Phi }{\partial y}}-{\frac {\partial G}{\partial t}}&=0{\text{,}}&-{\frac {\partial \Phi }{\partial z}}-{\frac {\partial H}{\partial t}}&=0{\text{.}}\end{aligned}}$ If $F$, $G$, $H$, $\Phi$ are the potentials of electromagnetic theory, these are precisely the expressions for the three components of magnetic force and the three components of electric force, given in the text-books. Thus the condition that distant intervals can be compared directly without specifying a particular route of comparison is that the electric and magnetic forces are zero in the intervening space and time.

It may be noted that, even when the coordinate system has been defined, the electromagnetic potentials are not unique in value; but arbitrary additions can be made provided these additions form a perfect differential. It is just this flexibility which in our geometrical theory appears in the form of the arbitrary choice of gauge-system. The electromagnetic forces on the other hand are independent of the gauge-system, which is eliminated by "curling."]

It thus appears that the four new quantities appearing in our extended geometry may actually be the four potentials of