CHAPTER XVIITHE MATHEMATICAL EXPRESSION OF EINSTEIN’S FUNDAMENTAL PREMISES
CONSIDER a Galilean system, and two pointsandin this system. If a ray of light is propagated fromto,the postulate of the invariant velocity of light demands that the distancemeasured by us in the frame, divided by the duration required (according to the time of our frame) for a light wave to cover this distance, be always equal to,the velocity of light. If, now, we had viewed the same phenomenon from some other Galilean system, the pointsandof the first system at the instants when the ray of light passed them would have corresponded to some other pointsandof our second system. But, just as in the first case, we should have been able to measure the velocity of light between the two pointsandof our second system. The principle of the invariant velocity of light states that in whatever Galilean system we might have operated, the measured velocity of lightin vacuowould always be the same.
Of course this velocity may be positive or negative, according to whether the light ray is directed to the right or to the left; but we can obviate this ambiguity of sign by considering squared values.
The mathematical translation of this principle of physics yields us the following equation, which must remain invariably zero in value for all Galilean frames:When we pass from one Galilean frame to another,,,,may vary in value; but the variations must be so connected that the sum total of the above mathematical expression remains invariably zero. Such is the mathematical condition which expresses the principle of the invariant velocity of light.
Now it is our object to determine exactly in what measure these different magnitudes,,,must vary in value when we pass from one Galilean frame to another, moving with a constant velocitywith respect to the first, if the previously written mathematical condition of invariance is to remain satisfied. From a purely mathematical standpoint problems of this type form part of a branch of mathematics known as the theory of invariants. Such problems had been studied many years before, and it was known that the relations between the variables,,,,or, what comes to the same thing, the transformations to which it was necessary to subject these variables (in order to satisfy the condition of invariance set forth above), were given by a wide group of transformations known asconformal transformations.[62]
But when, in addition, the relativity of velocity is taken into consideration it is seen thatconformal transformationsare far too general. We must restrict them; and when the required restrictions are imposed we find that the rules of transformation according to which the space and time co-ordinates of one Galilean observer are connected with those of another depend in a very simple way on the relative velocityexisting between the two systems. These rules of transformation are given by theEinstein-Lorentz transformations.[63]
Now these transformations are, as we have said, more restricted than the conformal transformations; and this lesser generality of the Einstein-Lorentz transformations has, as a consequence, the further restriction of the conditions of invariance of the mathematical expression mentioned previously. Not only will this expression have a zero value for all Galilean frames when it has a zero value for one particular Galilean frame, but in addition, if it does not happen to have a zero value in one frame but has some definite non-vanishing numerical value, it will still maintain this same definite non-vanishing value in all other Galilean frames. In other words, Einstein’s premises are represented mathematically by the invariance of the total value offor all Galilean frames, regardless of whether this value happens to be zero or non-vanishing.
The deep significance of this condition of invariance was first noted by Minkowski, and it led, as we shall explain in the next chapter, to the discovery of four-dimensional space-time.