Concepts of Simultaneity: From Antiquity to Einstein and Beyond

11 Symmetry and Transitivity of Simultaneity

As stated in chapter 7, in his 1905 paper on relativity, Einstein “assumed” that clock synchronization or simultaneity are symmetric and transitive relations ,1 and he defined the “time” of a reference system in terms of the notion of simultaneity. Because of the importance of the relation between these two notions, the “time of a reference system,” briefly denoted by t, and “simultaneity,” denoted by , it would be useful to recall the following settheoretical definitions. (I) A binary relation R on a set S is called “reflexive” if, for every element a of S, the proposition a R a is true, that is, if every element of S stands in relation R to itself; (II) R is said to be “symmetric” in S if, for any two elements a and b of S, a R b implies b R a; (III) R is said to be “transitive in S if, for any three elements a, b, c of S, a R b and b R c imply a R c; (IV) a relation R is said to be an “equivalence” relation if it is reflexive, symmetric, and transitive; (V) a collection of subsets of S is a “cover” of S if their (set-theoretical) sum is S; (VI) a cover of S is a “partition” of S if its members are pairwise disjoint. C H A P T E R E L E V E N Symmetry and Transitivity of Simultaneity 1 See chapter 7, note 34. 202 Concepts of Simultaneity With these definitions it is easy to prove the following theorem: Any equivalence relation in S leads to a unique partition of S and, conversely, any given partition of S defines an equivalence relation on S. If R is an equivalence relation on S and a any fixed (but arbitrary) element of S, then the set of all elements x of S that satisfy the condition x R a is called the equivalence class of a and denoted by [a]; hence, symbolically, [a]  {x⏐x  S and x R a} Finally, the collection of all equivalence classes generated in S by an equivalence relation R is called the quotient set of S modulo R (or induced by R) and denoted by S/R. If S denotes the set of all events, R the (standard) simultaneity relation , and t, the time of a reference system, then Einstein’s definition of time (Edt) as presented at the end of § 1 of his 1905 paper on relativity can be expressed symbolically by t  df S/ This definition says that the time of a reference system is the quotient set of all events induced by the (standard) simultaneity relation. True, Einstein never published such a set-theoretical formulation of his definition of time, but he would have undoubtedly endorsed it because it faithfully expresses what he had in mind. To substantiate this claim let us recall that in his Kyoto lecture, as mentioned in chapter 7, Einstein declared that “an analysis of time was my solution,” whereas he should have said “an analysis of (the concept of) simultaneity was my solution.” His referral to t, instead of to , suggests that he had Edt in his mind. This suggestion is strongly supported also by Wertheimer’s report of his discussions with Einstein , according to which Michelson’s famous experiment or other experimental discoveries did not lead to the genesis of the theory of relativity, but the fact that “it occurred to Einstein that time measurement involves simultaneity .” That Einstein early distinguished between distant simultaneity, as in Edt, and local simultaneity of events was also recorded by Wertheimer when he quoted Einstein as having said: “If two events occur in one place, I understand what simultaneity means. . . . But am I really clear about what simultaneity means when it refers to events in two places? What does it mean to say that this event occurred in my room at the same time as another event in some distant place? Surely I can use the concept of simultaneity for dif- ferent places in the same way as for one and the same place—but can I? Is it as clear to me in the former as it is the latter case? . . . It is not!”2 Einstein’s early awareness of the difference between local and distant simultaneity or consequently, in accordance with Edt, between local...