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From: "Paul.B.Andersen" <relativity@paulba.no>
Newsgroups: sci.physics.relativity
Subject: Re: The problem of relativistic synchronisation
Date: Mon, 2 Sep 2024 14:42:46 +0200
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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/