Page:BumsteadContraction.djvu/5

The distances $\scriptstyle {x}$, $\scriptstyle {y}$, $\scriptstyle {z}$ are measured by standards at rest, $\scriptstyle {\xi }$, $\scriptstyle {\eta }$, $\scriptstyle {\zeta }$, by standards in motion. The distance between two points (say on the $\scriptstyle {x}$-axis) when measured by the first is $\scriptstyle {x_{2}-x_{1}}$; when measured by the second, it is $\scriptstyle {\xi _{2}-\xi _{1}={\frac {x_{2}-x_{1}}{\sqrt {1-\beta ^{2}}}}}$. The length of the moving standards, when parallel to the axis of $\scriptstyle {x}$ are thus $\scriptstyle {\sqrt {1-\beta ^{2}}}$ times the fixed standards; when perpendicular to $\scriptstyle {x}$, they have the same length as the fixed standards. In order therefore that Einstein's principle should hold, it is necessary that all moving objects should suffer the Lorentz-FitzGerald contraction.

It is easy to compare the rates of the fixed and moving clocks by considering two events whose difference in time as measured by the fixed clocks is $\scriptstyle {t_{2}-t_{1}}$; as measured by a moving clock whose coordinate is $\scriptstyle {\xi ^{\prime }}$, let the interval be $\scriptstyle {\tau _{2}-\tau _{1}}$. Then

$\scriptstyle {\tau _{2}-\tau _{1}={\frac {1}{\sqrt {1-\beta ^{2}}}}\left[t_{2}-t_{1}-{\frac {v}{{\text{V}}^{2}}}(x_{2}-x_{1})\right]}$
But $\scriptstyle {x_{2}={\sqrt {1-\beta ^{2}}}\xi ^{\prime }-vt_{2}}$ and $\scriptstyle {x_{1}={\sqrt {1-\beta ^{2}}}\xi ^{\prime }-vt_{1}}$.
So that $\scriptstyle {\tau _{2}-\tau _{1}={\sqrt {1-\beta ^{2}}}(t_{2}-t_{1})}$.

Thus the moving clocks run slower than the fixed ones; if a clock at rest beats seconds, it must, when in motion, have a period of $\scriptstyle {\frac {1}{\sqrt {1-\beta ^{2}}}}$ seconds.^[1]

It is possible that the principle of relativity may come to be regarded as one of the fundamental empirical laws of Physics, occupying a position analogous to that of the Second Law of Thermodynamics. It rests on a similar basis, in that no deviations from it have been observed. Indeed the analogy may be made more complete by showing that the denial of the principle leads to a third kind of perpetual motion, by which the kinetic energy of any body might be exhausted and the body be brought to rest with reference to the ether.^[2] There is however an enormous difference in the breadth of the evidence on which the two principles rest. Violations of the principle of relativity lead only to minute effects which must be sought for in difficult and recondite experiments. The fact remains however that, so far as our knowledge extends, the principle holds; the most reasonable course in regard to it, and that which promises to be most fertile in results, is to accept it provisionally and to develop its conse-

1. This relation between the time in fixed and moving systems was also taken into account by Lorentz, by means of a variable which he calls local time. Versuch einer Theorie, §31.
2. Larmor in FitzGerald's Papers, p. 566.