Re: The problem of relativistic synchronisation

Den 01.09.2024 22:29, skrev Richard Hachel:

OK, Paul.
 So we're going to continue, because it's very important.

OK. Let's play.
https://paulba.no/pdf/Mutual_time_dilation.pdf
c ≈ 3e8 m/s
d = 3e8 m
v = 0.8c
tA(e1) = 0
tA'(e1)= 0
tA(e2) = (d/v)⋅√(1−v²/c²) ≈ 0.75 seconds.

 A sees the segment AB coming towards him, and when A' crosses A, which is event e1, A starts his watch. tA(e1)=0
 Then A observes that B' is approaching him at high speed, and stops his watch when B crosses him, this is event e2, and we note tA(e2)=0.75.
 There is an interval of 0.75 seconds between e1 and e2.
 For 0.75 seconds, B' rushes toward A.
 1. At what apparent (i.e. APPARENT) speed does A apprehend B' rushing toward him?

I have no idea of what your "apparent speed" is, and I don't care.
I can however tell you what A will _measure_ the speed of B'
relative to A will be.
A knows that his clock shows 0 at e1, when B' is at
the position x = (-3e8m + (v/c²)⋅0)/√(1−v²/c²) = -3e8m/√(1−v²/c²)
A knows that his clock shows 0.75 seconds when B' is at
the position x = 0.
So B' has moved the distance 3e8m/√(1−v²/c²) in 0.75 s
v = (3e8m/0.75s)/√(1−v²/c²)
v = 3e8m/√((0.75s)²+(3e8m)²/(3e8m/s)²) = 240000000 m/s = 0.8c

 2. What is the apparent distance traveled by B' during the interval noted by the watch?

The distance B' has travelled in the rest frame of A is shown above.
d' = 3e8m/√(1−v²/c²) = 500000000 m
It is nothing 'apparent' about this distance.
Now that I have answered your questions, you can possibly answer mine:
When the train leaves Nantes, you see the watch on the railway station
showing 8:32, and you start your stop watch. When you arrive in Berlin,
you see the watch on the railway station showing 20:41. You stop your
stop watch which show that the duration of the journey was 12h 9m.
The question:
-------------
Why is the difference between the Berlin clock at arrival and
the Nantes clock at departure equal to the duration of your journey,
  20:41 - 8:32 = 12h 9m ?
--
Paul
https://paulba.no/