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to synchronize two clocks. Moreover, according to this hypoth- esis, the phenomena of nature go on just the same whatever the value of u, so that the want of synchrony cannot in any way show itself in fact, if it did, it would immediately become possible to measure the effect and so arrange for true synchrony.

As the earth moves in its orbit, the value of u changes, so that its value in the spring, for instance, will be different from its value in the autumn. One pair of astronomers may attempt to synchronize a pair of clocks in the spring, but their synchro- nization will appear faulty to a second pair who repeat the deter- mination in the autumn. There will, so to speak, be one syn- chrony for the spring and another for the autumn, and neither pair of astronomers will be able to claim that their results are more accurate than those of their colleagues. More generally we may say that different conceptions of synchrony will cor- respond to different velocities of translation.

These elementary considerations bring us to the heart of the problem which we illustrated diagrammatically in fig. 2. The observer at O in the diagram will have one conception of simultaneity, while the second observer who moves from O to P will, on account of his different velocity, have a different con- ception of simultaneity. The instants at which the wave front of the light signal from O reaches the points A, B, C in the diagram will be deemed to be simultaneous by the observer who remains at O, but the observer who moves fron O to P will quite un- consciously have different ideas asto simultaneity. At instants which he regards as simultaneous the wave front will have some form other than that of the sphere ABC surrounding O. If the hypothesis of relativity is to be true in its application to the transmission of light signals, this wave front must be a sphere having P as its centre.

Einstein examined mathematically the conditions that this should be possible. Unfortunately a precise statement of his conclusions can only be given in mathematical language.

The observer who is supposed to remain at O in fig. 2 may be supposed to make exact observations and to record these observations in mathematical terms. To fix the positions of points in space he will map out a " frame of reference " consisting of three orthogonal axes, and use Cartesian coordinates x, y, z, to specify the projections along these axes of the radius from the origin to any given point. He will also use a time coordinate t which may be supposed to specify the time which has lapsed since a given instant, as measured by a clock in his possession. Any observations he may make on the transmission of light signals can be recorded in the form of equations between the four coordinates x, y, z, t. For instance, the circumstance that light travels from the origin with the same velocity c in all directions will be expressed by the equation (of the wave front):

The second observer who moves from O to P will also construct a frame of reference, and we can simplify the problem by supposing that his axes are parallel to those already selected by the first observer. His coordinates, to distinguish them from those used by the first observer, may be denoted by the accented letters x', y', z', t'. If his observations also are to show light always to travel with the same velocity c in all directions, the equation of the wave front, as observed by him, must be:


 * ' 2 +y 2 +2 /2 -^'=o. 2 ... (2)

A 19th-century mathematician would have insisted that x, y, z, t must be connected with x', y', z', t' by the simple rela- tions:

x' = x u t

y'=y

Z'=Z

t'=t

(A)

but it is obvious that if these relations hold, then equation (i) cannot transform into equation (2). Einstein finds that equation t (i) will transform into equation (2) provided the coordinates
 * , y, z, t of the first observer are connected with the coordinates
 * ', y', z', t' of the second observer by the equations:

y'=y

z'=z

(x-ut) 1

(B)

/ ^ ^ where j3 stands for ( I ^?

To form some idea of the physical meaning of these equations, it will be advantageous to consider the simple case in which the first observer is at rest in the aether while the second moves through the aether with velocity u. The points of difference between equations (B) and (A) then admit of simple explanation. The factor /3 in the first of equations (B) is simply, according to the suggestion of Fitzgerald and Lorentz already mentioned, the factor according to which all lengths parallel to the axis of x must be adjusted on account of motion through the aether with velocity. The moving observer must correct his lengths by this factor, and he must correct his times by the same factor in order that the velocity of propagation of light along the axis of x may still have the same velocity c; this explains this presence of the multiplier /3 in the last of equations (B). The one remain- ing difference between the two sets of equations, namely the

replacement of t in (A) by / - in (B), represents exactly the

C

want of synchrony which, as we have already seen, is t be expected in the observations of two observers whose velocity differs by a velocity.

Although the equations admit of simple illustration by con- sidering the case in which one observer is at rest in a supposed aether, it will be understood that the equations are more general than the illustration. They are in no way concerned with the possibility of an observer being at rest in an aether, or indeed with the existence of an aether at all. Their general interpretation is this: If one observer O, having any molion whatever, finds, as a matter of observation, that light for him travels uniformly in all directions with a constant velocity c, then a second observer P, moving relative to O with a constant velocity M along the axis of x, will find, as a matter of observation, that light, for him also, travels uniformly in all directions with the same constant velocity c, provided he uses, for his observations, coordinates which are connected with the coordi- nates of O by equations (B).

This is the meaning that was attached to the equations by Einstein in 1905, but the equations had been familiar to mathe- maticians before this date. They had in fact been discovered by Lorentz in 1895 as expressing the condition that all electro- magnetic phenomena, including of course the propagation of light, should be the same for an observer moving through the aether with velocity u as for an observer at rest in the aether. For this reason the transformation of coordinates specified by these equations is universally spoken of as a " Lorentz transformation." What Einstein introduced in 1905 was not a new system of equations but a new interpretation of old equations. The two obs0vers who used the coordinates x, y, z, t and x 1 ', y', z', I' had been regarded by Lorentz as being one at rest in an aether and one in motion with a velocity u; for Einstein they were observers moving with any velocities what- ever subject to their relative velocity being u. Lorentz had regarded t as the true time and t' as an artificial time. If the observer could be persuaded to measure time in this artificial way, setting his clocks wrong to begin with and then making them gain or lose permanently, the effect of his supposed artifici- ality would just counterbalance the effects of his motion through the aether. With Einstein came the conception that both times, / and /', had precisely equal rights to be regarded as the true time. The measure t' is precisely that which would be adopted naturally by any set of observers, or race of men, who disregarded their steady motion through space; their adoption of it would be above criticism if, as Einstein suggested, their motion through space had no influence on material phenomena, and it represents, as we have seen, the usual practice of astronomers in comparing time at different places. From this point of view,