There are two frames F1 and F2 where F2 has a speed v in x direction with respect to F1. There are three observers A,B and C where at time 0, A&B are in the frame F1 and C in the frame F2. B stands at location 0. The following events are all occured in the x direction. The positions and speeds are zero in all the other directions.

Events in the F1 frame

==================

Event e(-s,0)

---------------
Any event occured at place xe=-s and time te=0. For example emitting a light pulse from a light source

Event B:as(0,0)

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Observer B starts accelerating in x direction.
position xas= 0, time tas=0, speed = 0

Event B:af(xaf,taf)

--------------------

Observer B stops accelerating when they reach the speed v. After this time they are in the frame F2.
location xaf=xaf, time taf=taf

A diagram:
-------------------------

direction x ->

events.......e.................B:as.....B:af
locations...-s===================0=======xaf
times........0...................0.......taf

Events in the F2 frame

======================

The same events occurs in all the reference frames. They are all occured in the x direction. The positions and speeds are zero in all the other directions. The speed of the F1 is -v in the x direction. So , for the frame F2, the following transfer functions can be used. (x',t') and (x,t) are the coordinates in the frames F1 and F2 respectively.
x' = d.(x - v.t)
t' = d.(-x.v/c^2 + t)

where d=1/sqrt(1-v^2/c^2),

event e(xe',te') where xe=-s,te=0
----------------------------------

xe' = d.(xe - v.te) = d.(-s - 0)

xe' = d.(-s)

te' = d.(-xe.v/c^2 + te)=d.(-(-s).v/c^2+0)

te' = d.s.v/c^2

event B:as(xas',tas') where xas=0,tas=0

--------------------------------------

Observer B starts accelerating in x direction.
xas' = d.(xas -v.tas) = 0

tas' = d.(-xas.v/c^2 +tas)=0

event B:af(xaf',taf') where xaf=xaf,taf=taf

---------------------------------------------

Observer B stops accelerating with speed 0. They are in this frame from now on.
xaf' = d.(xaf-v.taf)

taf' = d.(-xaf.v/c^2+taf)

A diagram:
-------------------------------------------

direction x ->

events.......e...................B:as.....B:af

positions...xe'===================0=======xaf'

times.......te'...................0.......taf'

An interesting thing here is that the time te' which is the event's time has a positive value. The values s,xas and tas are all independent so the time te' can be greater than the time tas' which is the time when the acceleration stops.

te' > tas'
d.(s.v/c^2) > d.(-xaf.v/c^2+taf)
(s.v/c^2) > (-xaf.v/c^2+taf)
s > (-xaf+taf.c^2/v)

Events in the observers B's frame
==================================
Initially B is in the frame F1. At time 0, an event occurs at location -s. they also starts accelerating at this time and whenever they reach the frame F2 they stops accelerating. Reaching time of the frame F2 is taf' with respect to this frame. However if s is sufficiently large, the event's time te' would be greater than this time and thus the event would occur after the observer B has reached the frame F2.

Obviously this can not occur in a universe where time advances always even in accelerated frames. For example if the event is the emitting of a light pulse, this light pulse will travel some distance during the acceleration. However, after the acceleration phase, the observer would found that the ligh pulse is not emitted yet.

On Monday, 13 March 2023 at 12:30:16 UTC+1, Ufuk Denizyar wrote:
> There are two frames F1 and F2 where F2 has a speed v in x direction with respect to F1. There are three observers A,B and C

Do these observers have their own brains? Do they
have the ability of deciding what to say themself - or are
YOU deciding for them?

Who thought of that?