Page:TolmanFundamental.djvu/4

Rh The substitutions

$\dot{u}_{x}'=\frac{\dot{u}_{x}}{\left(1-\beta^{2}\right)^{\frac{3}{2}}},\ \dot{u}_{y}'=\frac{\dot{u}_{y}}{\left(1-\beta^{2}\right)}\ \mathrm{and}\ \dot{u}_{z}'=m_{0}\frac{\dot{u}_{z}}{\left(1-\beta^{2}\right)}$|undefined

are an obvious consequence of the relations between the units of length and time used in the two systems. For example, if a body has an acceleration in the Y direction, of magnitude $$\dot{u}_{y}$$when measured in the system S, evidently its acceleration $$\dot{u}_{y}'$$ as measured in the system S' will be greater because the units of time used in that system are "lengthened" in the ratio $$1:\sqrt{1-\beta^{2}}$$. Remembering that the units of length in the Y direction are the same in both systems, and noticing the time enters to the second power in the expression for acceleration, the relation $$\dot{u}_{y}'=\tfrac{\dot{u}_{y}}{\left(1-\beta^{2}\right)}$$ is evident. The other relations may be obtained in a similar way.

If now we define force as the increase in momentum per second we shall have, as has already been pointed out by Lewis ,

$\mathsf{F}=\frac{d}{dt}(mu)=m\frac{du}{dt}+u\frac{dm}{dt},$

where a possible change in mass as well as a change in velocity is allowed for. It has, moreover, been shown by Professor Lewis and the writer, that the two postulates of relativity, themselves, combined simply with the principle of the conservation of momentum are sufficient for a proof that the mass of a body is increased when set in motion in the ratio $$1:\sqrt{1-\beta^{2}}$$, so that in general the mass of moving body $$m=\tfrac{m_{0}}{\left(1-\beta^{2}\right)}$$. Substituting in the equation above, we have

$\mathsf{F}=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{du}{dt}+u\frac{d}{dt}\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}},$|undefined