Page:TolmanFundamental.djvu/4

The substitutions

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

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^[1],

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

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^[2], 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

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

1. Lewis, Phil. Mag. xvi. p. 705 (1908).
2. Lewis and Tolman, loc. cit.