232 ANNUAL REPORT SMITHSONIAN INSTITUTION, 19 3 3 



provided the clocks were synchronized to read zero at the instant 

 when A and B parted company. In words, B assigns to an event 

 near himself an earlier epoch, by his clock, than A assigns to the 

 same event by his clock. Similarly, A assigns to an event at himself 

 an earlier epoch than B assigns to the same event. To A, B's clock 

 runs slow; to B, A's clock runs slow. The degree of running slow is 

 the more pronounced the faster the relative motion. 



This is a well-known result in the theory of relativity,^ but our 

 presentation of it has a different logical setting. We have not used 

 the concept of a rigid measuring rod, and we have not assumed that 

 A and B are in " empty space." We have not taken the constancy of 

 the velocity of light from observation. We have adopted it as an 

 axiom giving us our definitions of lengths. We have simply assumed 

 that A and B possess temporal experiences and that they stand in 

 completely equivalent relationships to one another. This condition 

 is satisfied if A and B stand in the same relation to the rest of the 

 universe. Our experimental basis is that it is actually possible to 

 realize in experience two equivalent observers in uniform relative 

 velocity. 



Let us in fact apply this result to the whole universe. It is well 

 known to everybody nowadays that the universe as a whole is 

 expanding. By this we mean that the extragalactic nebulae are 

 receding from us and from one another, receding from one another, 

 moreover, at rates which are proportional to their separations. 



This is the same thing as saying that if we take any nebula and 

 divide its velocity by its distance we get the same result for all 

 nebulae. We do not know from observation how this quotient 

 changes in time. It appears to be the same all over the universe at 

 the present epoch (i.e., for nebulae considered simultaneously in the 

 above sense of simultaneity) as far as is yet observed. Whether at 

 another epoch it will be the same we do not know. Some current 

 theories assume that the coefficient of the proportionality is locally 

 constant in time, and hence that the nebulae are being accelerated. 

 Now, however a nebula is being accelerated, its velocity can never 

 exceed the velocity of light.* Thus, if the nebulae were being accel- 

 erated outward, each one would ultimately acquire the velocity of 

 light, and the law of proportionality of distance and velocity would 

 cease to be obeyed, for the velocities would tend to become the same 

 for all nebulae, but the distances would be different for different 



^ What are usually called the " Lorentz formulae ", connecting A's and B's assignments 

 of epochs and coordinates to any event whatever, can be shown to follow from the above 

 formulae without further assumptions. 



* Statements sometimes made to the contrary, as for example that nebulae may ulti- 

 mately possess velocities greater than that of light and so pass out of causal connection 

 with the rest of the universe, are erroneous. The error came in owing to a false identifi- 

 cation of " cosmic " time with the time of experiemce. 



