ESSAYS 653 



replaced by -v, the velocity of 5 relative to S'. Thus the relation between S 

 and S' is entirely reciprocal, so that neither occupies a special position, tho 

 consideration of rest or motion in the ether being eliminated, and replaced by the 

 relative uniform velocity of either system to the other. 



Einstein's General Principle of Relativity may be expressed in the form : The 

 mathematical expressions of the laws of physical phenomena are covariant with 

 respect to the Lorentz transformation. Einstein has shown that this condition is 

 rigorously fulfilled by Maxwell's equations for stationary media, the small deviation 

 in the presence of electric charges in Lorentz's representation being now 

 eliminated. 



We cannot say, in this method of representation, that a longitudinal length 

 L' in S' moving with velocity v relative to 5 is shortened by its relative motion, 

 but that it will appear shortened to L'/y to observers in 5" observing its two ends 

 simultaneously. A phenomenon in S', of duration T to S' observers, will appear 

 to S observers to have its duration increased to y7". 



The starting point of Minkowski's representation was his finding that the 

 Lorentz transformation might be expressed in the form .r' = ^rcosco + ;sin&>, y' =y, 

 z 1 = z, /' = /cos» — jrsino), where / = v/ - 1 ct a ict, say, /' = ict', &> = tan " V/3. Now 

 this is a pure rotation, in the four- dimensional space x, y, z, ict, through an 

 imaginary angle a> in the plane x, ict, and therefore round the plane yz (i.e. the 

 whole plane yz remains fixed, just as in three dimensions a rotation is round a 

 fixed line, the axis). There is, therefore, no change in the length </x 2 +y 2 + z 2 - c 2 t 2 , 

 so that its square, x 2 +y 2 + z 2 -c 2 t 2 , will retain its value unchanged, i.e. both are 

 invariants of the transformation. But/ and z being unchanged, so will /* + .?*, and 

 hence also x 2 — c 2 t 2 . Now, the whole history of a moving particle or light-ray in 

 space and time will be represented in the Minkowski space-time, or world, by a 

 continuous line of point instants, which he calls a world-l'me. 



Take, after Minkowski, the case of a light-ray, or particle moving with the 

 velocity of light, along the .r-axis, so that we have to consider only the plane 

 section of the world containing the .r-axis and time-axis. We shall also take the 

 world defined by x, y, z, ct, in which the Euclidian rotation through the imaginary 

 angle o> in x,y, z, ict will become a real non-Euclidian rotation through the angle 

 ^ = tan - J /3. Through a point O as origin draw the .r-axis SON (south to north) and 

 the ct-Rxis WOE, ON, OS, OE, and W being all equal, and representing the units 

 ^= + iandc/= + i respectively. Draw the two bisectors S W O NE and NIVO SE 

 of the right angles NOE and NO W. These will be the asymptotes of the two con- 

 jugate hyperbolae x 2 - c 2 t 2 = ± I. The line S W NE will represent the world-line of 

 the ray or particle. To represent either as at rest, i.e. to pass from the system S to 

 the system S' moving with the ray or particle, turn CWand OE through the angle 

 ^ = tan-!/3 and both towards, or both away from, the asymptote ONE, anA if N' and 

 E' are the points in which they intersect the conjugate hyperbola?, the lines ON' 

 and OE' are the new axes and their lengths are the new units in which x' and ct 

 are measured. Since x 2 -c 2 t 2 is invariant, the asymptotes (x 2 -c 2 l 2 = o) and the 

 hyperbola? (x 2 -c 2 t 2 = ± 1) remain fixed, as would be the case for the whole system 

 of conjugate hyperbola? x 2 -c 2 t 2 = ±f 2 , where « is any real number. Since the 

 velocity v can never exceed c, world-lines through O must clearly be confined to 

 the region included by the angles NE SE and NIV SW. Since a time-axis 

 can be drawn from O through any world-point for which x 2 -c 2 t 2 <o, and an 

 x-axis through any for which x 2 - c 2 t 2 >o, it follows that any point on any world-line 

 can be made simultaneous to 0, or any light-ray or moving particle transformed to 

 rest. The case of motion in a plane requires a three-dimensional world for its 



