Suppose that two observatories, say Greenwich and Paris, wish to synchronize their clocks, with a view to, let us say, an exact determination of their longitude difference. Paris will send out a wireless signal at exact midnight as shown by the Paris clock, and Greenwich will note the time shown by the Greenwich clock at the instant of receipt of the signal. Greenwich will not, however, ad just their clock so as to show exact midnight when the signal is received; a correction of about .00i second must be made to allow for the time occupied by the signal in traversing the distance from Paris to Greenwich. To turn to mathematical symbols, if is the time at which a signal is sent out from one station, the time of receipt at a second station is taken to be where x is their dis tance apart, and c is the velocity of light. This represents the ordi nary practice of astronomers, but it is clear that if the earth is travelling through a fixed ether with a velocity u in the direction of the line joining the two observatories, the velocity of transmis sion of the signal relative to the two observatories will not be c but C-1-11, and the time of receipt at the second station will be c+u Thus it appears that it is impossible to synchronize two clocks unless we know the value of u, and that the ordinary practice of astronomers will not, as they expect, synchronize their clocks, but set them at an interval apart equal to According to the hypothesis of relativity, it is impossible ever to determine the value of u, and so it is impossible ever truly to synchronize two clocks. Moreover, according to this hypothesis, 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 meas ure 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 synchro nize a pair of clocks in the spring, but their synchronization will appear faulty to a second pair who repeat the determination in the autumn. There will, so to speak, be one synchrony 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. Different conceptions of synchrony will corre spond to different velocities of translation.
These elementary considerations bring us to the heart of the problem which we illustrated diagrammatically in fig. 2. The ob server at 0 in the diagram will have one conception of simul taneity, while the second observer who moves from 0 to P will, on account of his different velocity, have a different concep tion of simultaneity. The instants at which the wave front of the light signal from 0 reaches the various points A, B, C in the diagram will be deemed to be simultaneous by the observer who remains at 0, but the observer who moves from 0 to P will quite unconsciously have different ideas as to simultaneity. At instants which he regards as simultaneous the wave front will have some form other than that of the sphere ABC surrounding 0. 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. A precise statement of his conclusions can only be given in mathematical language.
The observer who remains at 0 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 co-ordinates x, y, z, to specify the projec tions along these axes of the radius from the origin to any given point. He will also use a time co-ordinate t which may be supposed to specify the time which has lapsed since a given instant, as meas ured 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 co-ordinates 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 0 to P will also construct a frame of reference, and we can simplify the problem by suppos ing that his axes are parallel to those already selected by the first observer. His co-ordinates, 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 A 19th-century mathematician would have insisted that x, y, z, t must necessarily be connected with x', y', z', t' by the simple rela tions:— Indeed, he would have been unable to imagine that there should be any other relation connecting these quantities. It is, however, ob vious that if these relations hold, then equation (I) cannot trans form into equation (2). Einstein showed that equation (I) will transform into equation (2) provided the co-ordinates x, y, a, t of the first observer are connected with the co-ordinates x', y', z', t' of the second observer by the equations :— To form some idea of the physical meaning of these equations, let us consider the simple case in which the first observer is at rest in the ether while the second moves through the ether with velocity a. The points of difference between equations (B) and (A) then admit of simple explanation. The factor 13 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 ether with velocity a. 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 the presence of the multiplier 3 in the last of equations (B). The one remaining difference between the two sets of equa tions, namely, the replacement of t in (A) byin (B), rep.