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XXXV. Note on the Derivation from the Principle of Relativity of the Fifth Fundamental Equation of the Maxwell-Lorentz Theory.

By Richard C. Tolman, Ph. D., Instructor in Physical Chemistry at the University of Michigan.^[1]

If we consider two systems of "space time coordinates” S and S' in relative motion in the X direction with the velocity v, any kinematic phenomenon which occurs may be described in terms of the variables x, y, z and t belonging to the system S or z', y', z' and t' belonging to the system S'. The Einstein theory of relativity has led to the following equations for transforming the description of a kinematic phenomenon from one set of coordinates to the other^[2].

$t'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left(t-{\frac {v}{c^{2}}}x\right)$

(1)

$x'={\frac {1}{\sqrt {1-\beta ^{2}}}}(x-vt)$

(2)

$y'=y\,$

(3)

$z'=z\,$

(4)

(where $c$ is the velocity of light and $\beta$ is substituted for the fraction ${\tfrac {v}{c}}$).

The content of these equations may be expressed in words, by saying that an observer in the moving system S' (S having been arbitrarily taken as at rest) uses a metre stick which, although the same length as a stationary metre stick when held perpendicular to the line of relative motion of the two systems, is shortened in the ratio of ${\sqrt {1-\beta ^{2}}}:1$ when held parallel to OX, that clocks in the moving system beat elf seconds which are longer than those of stationary clocks in the ratio $1:{\sqrt {1-\beta ^{2}}}$, and that a clock in the moving system which is $x'$ units to the rear of the one at the centre of coordinates is set ahead by $x'{\tfrac {v}{c^{2}}}$ seconds, although the two clocks appear synchronous to the moving observer. A simple non-analytical derivation of these relations has been given in another place^[3].

Let us now take the Maxwell-Hertz equations for the

1. Communicated by the Author.
2. Einstein, Arm. d. Physik, xvii. p. 891 (1905); Jahrbuch der Radioaktivität, iv. p. 411 (1907).
3. Lewis and Tolman, Proc. Amer. Acad. xliv. p. 711 (1909); Phil. Mag. xviii. p. 510 (1909).