Page:TolmanFundamental.djvu/3

${\mathsf {H}}_{y}'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left({\mathsf {H}}_{y}+{\frac {v}{c}}{\mathsf {E}}_{z}\right)$

(17)

${\mathsf {H}}_{z}'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left({\mathsf {H}}_{z}-{\frac {v}{c}}{\mathsf {E}}_{y}\right)$

(18)

Thus at a given point in space, we may distinguish between the electric vector E as measured by a stationary observer and the vector E as measured in units of his own system by an observer who is moving past the stationary system with the velocity $v$ in the X direction. If $\epsilon$E is the force acting on a small stationary test charge of magnitude $\epsilon$, then $\epsilon$E' will be the force acting on the same test charge or electron when it is moving through the point in question with the velocity $v$, the force $\epsilon$E' being measured in units of the moving system^[1].

We are more particularly interested, however, in the vector F which determines the force $\epsilon$F that nets on the moving charge but which is measured in "stationary units," thus determining the equations of motion of the test charge $\epsilon$ with respect to stationary coordinates. Since, however, it is possible to obtain relations between the units of force used by stationary and moving observers, a method is presented of calculating F from the values of E' already given by the transformation equations (13-18). As a matter of fact the expression for F which can thus be obtained is identical with the fifth fundamental equation of the Maxwell-Lorentz theory.

Relation between the Units of Force used in Moving and Stationary Systems.

Consider a body having; the mass $m_{0}$ when at rest and moving with the same velocity $v$ as a system of coordinates S'. Evidently its acceleration with respect to those coordinates is determined by Newton's laws of motion, and its acceleration with respect to stationary coordinates can be found by making the proper substitutions, giving us

${\mathsf {F}}_{x}'=m_{0}{\dot {u}}_{x}'=m_{0}{\frac {{\dot {u}}_{x}}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}$

(19)

${\mathsf {F}}_{y}'=m_{0}{\dot {u}}_{y}'=m_{0}{\frac {{\dot {u}}_{y}}{\left(1-\beta ^{2}\right)}}$

(20)

${\mathsf {F}}_{z}'=m_{0}{\dot {u}}_{z}'=m_{0}{\frac {{\dot {u}}_{z}}{\left(1-\beta ^{2}\right)}}$

(21)

1. It should be noticed that according to the first postulate of relativity, if the charge of a stationary electron, for example a hydrogen ion, is $\epsilon$, then when the electron is in motion it must still appear to have the charge $\epsilon$ to an observer who is moving along with it, otherwise the possibility would be presented of distinguishing between relative and absolute motion. This justifies us in taking $\epsilon$E' as the force acting on the moving electron and measured in the moving system.