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**Extra resources for Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time**

**Example text**

Given particles Q,R,S and an instant Rx £ R such that Ir 1 (R RQ x ), R , f (R ) > and x 8R x Ir R8 24 1 (R ). 9] we say that the instants R x is between the particles Q and S. ~ > . S _ _ and instants Qc E Q. _ S c E S such that Qc ~ Se' there exists a particle ~ such that That is, there is a particle between distinct from both. 1). 9 The Isotropy of SPRAYs Any set of particles which coincide simultaneously at a given event is called a SPRAY. RE(J>}. That is, SPR(Qa] is the set of particles which coincide (with Q) at the event [Qa] (see Fig.

The is a unique extension of 0 on there are extended signal funations, relation, a on ! events, an extended a to o. Similarly, an extended aoinoidenoe re~ation of "in optiaa~ line", and an extended betweenness relation, all of whiah be deno ted by the previous symbo ls. The (extended) wil~ signa~ funotions are bijeations whiah send ordinary instants onto ordinary instants and ideal instants onto ideal instants. following theorem is essentially the same as a result of Walker [1948, P 322J. THEOREM 8 (Extended Signal Functions) Then (i) f SQ (ii) f SQ ~ f SR 0 f RQ .

X y (i) If Qx Then (J (ii) If Qx a E y Rx (J (J ... 2). 2). 4) PROOF. 2) by induction on the number of particles. part (i) by Theorem 8(i). 4 Part (ii) follows from 0 Particles Do Not Have First or Last Instants The next theorem applies to ideal as well as to ordinary instants. 1, P323]. THEOREM 11. Particles do not have first or last instants. 2). 3).