- #106

PeterDonis

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Freixas said:when B and A are together and agree that their clocks are synchronized at 0, BN disagrees and says that A's clock actually reads 3.2.

No. Both B1 and BN are at rest relative to each other, so they have the same notion of simultaneity. So whatever B1 says A's clock reads at a given time, BN will say the same. That's what "clock synchronization" means.

To put it another way: the purpose of having the whole fleet of B observers is so that, at any event of interest, there will be some B observer there to record his clock time. At the event where B1 meets A, that B observer is, of course, B1. But the definition of "time" in the B frame just

*is*whatever time is recorded on the clock of the B observer who is located at that event. So B1's clock time--zero--when he meets A is also the "time" at which every other B observer says that event happened.

Yet another way of looking at it: A's clock reading at the event where B1 meets A is an invariant--it's 0. That's true for all observers in all frames (you've already agreed to this). So every B observer must say that,

*at the event at which B1 meets A*(which is also an invariant--it's an event), A's clock reads 0. But by the definition of "time" for all B observers, all B observers say that that event happened at time zero. That's what "time" in the B frame

*means*. So every B observer must say that, at his time 0, A's clock reads 0.

What every B observer will

*not*say is that, at his time 0,

*the time on the clock of the A observer who happens to be passing him at that instant*will read zero. That, of course, is false--we already know a counterexample, since we know that when BX passes M, BX's clock reads zero, but M's clock reads 3.2. But that does not mean that BX will say that, when he passes M,

*A's clock*reads 3.2. Why not? Relativity of simutaneity! (But correctly applied this time.) Or, to put it another way, the A clocks are offset in the B frame--A's clock runs 3.2 years earlier than M's clock does. So BX can calculate, from that clock offset, that when he passes M, and M's clock reads 3.2, A's clock must read zero. And of course he gets that confirmed when he gets the message from B1 that, when B1 passed A (which happened at clock time 0 for B1, so for BX, it happened at the same time BX passed M), A's clock read 0.