THE LOGIC OF RELATIVITY
Postulate V. If the velocity of a system of reference S% relative to a system of reference Si is measured by means of the units belonging to Si and if the velocity of Si relative to St is measured by means of the units belonging to St the two results will agree in numerical value.
Postulate T. If two systems of referencemove with unaccelerated
relative velocity and if a body moves relatively to one of the systems in a straight line with unaccelerated velocity then it also moves in a straight line relatively to the other and with unaccelerated velocity.
Postulate L. If two systems of reference Si and Si move with unaccelerated relative velocity and if a line segment I is perpendicular to the line of relative motion of Si and S$ and is fixed to one of these systems, then the length of I measured by means of the units belonging to Si will be the same as its length measured by means of the units belonging to S2.
We now Lave before us the logical basis upon which may be built the restricted theory of relativity in all its details. It has been put in essentially the same form as that employed in my "Theory of Relativity" (published by Wiley and Sons, New York) and in my earlier articles in "The Physical Review". Reference may be made to the book named for the detailed development of the theory. Here we shall attempt to sketch only the progress of ideas and to indicate the main conclusions.
The first thing to be done in developing the theory on this basis is to consider carefully the relation between the time units of the two systems. The following remarkable conclusion is reached by a process of reasoning which is fully cogent in character:
If two systems of reference Si and 52 move with a relative velocity v and is the ratio v/c of v to the velocity c of light as measured on either system, then to an observer on Si the time unit of Si appears to be in the ratio to that which is described to him as a unit by an observer on 52 while to an observer on 52 the time unit of 52 appears to be in the ratio to that
which is described to him as a unit by the observer on Si.
Thus we have the extraordinary conclusion that the time units of the two systems of reference Si and 52, not at rest relatively to each other, are of different lengths in such a way that an observer on either system thinks that the time unit of the other system is greater than his own. It is evident that no simple change of the unit on either system (or both) will bring the units into agreement for observers on both systems. As postulates V and L and T are generally accepted and have not elsewhere led to such strange conclusions it is natural to suppose that the strangeness here is not due to them. In the argument the restricted principle of relativity needs to be used only in so far as it is involved in the conclusion that the units of any two systems of reference 5i and S% are such that the same numerical result is obtained in measuring with the units of 5i a quantity L\ and with the units of 52 a quantity L% when the relation of L\ to 5i is precisely the same as the relation of 2 to 52. But this principle is accepted in the classical mechanics and has not elsewhere led to strange results. The conclusion in postulate M appears to be demanded