PHILOSOPHY OF PHYSICS MILNE 



231 



B's clock. It is easy to show that the direct reading by A through a 

 telescope is equivalent to the communication of information by B 

 and that the signal velocity is equal to the arbitrarily chosen 

 number c. 



It by no means follows that A and B agree on the epoch they 

 assign to the event which consists of reading B's clock. This was 

 Einstein's great discovery. A procedure for correlating the read- 

 ings, equivalent in the sequel to Einstein's results, is as follows: Con- 

 sider an event at B to which B assigns the epoch t'n (by direct read- 

 ing of his clock) ; and let A assign the epoch tn to it by his clock 

 after making the necessary light signals. Consider, secondly, an 

 event Ea at A, to which A assigns the epoch ^a (by direct reading of 



Suffixes A and B denote loc.itions of events. Unprimed symbols {(a, fe) are A's 

 assignments of epoch. Primed symbols (t'A, t'B) are B's assignments of epoch 



A's clock) ; and let B assign the epoch t\ to Ea by his clock after 

 making the necessary signals. Then, since A and B are supposed to 

 be completely equivalent in all their relationships, if t'-^ happens to 

 equal ^a, tn must equal t\. In other words, suppose that A makes a 

 graph of his value for the epoch of Eb, namely, ^'u against B's value 

 for the same event, namely f^i ; then the same graph must result if 

 B plots t\^ his value for the epoch of Ea, against t,-^-, A's value. These 

 graphs could be actually drawn and compared. In mathematical 

 language we should have t\=f{t^) and ^a=^/('^'a), where the two /'s 

 denote the same function.^ 



Now, it can be shown mathematically that the only possible form 

 of the graph for which this relation holds is represented by 



and 



^a=^'a(I-VVc^)v. 



* When A and B have chosen clock prnidimtions for which the two fs are Identical, they 

 may be Bald to be provided with identical clocks. 



