This is a fanciful little scenario for 'flexing the muscles' of acceleration in spacepropertime a little and so giving a better intuitive feel for this representation of relativity. Remember, this is not some 'new theory', just an alternative view of standard Einstein Special Relativity.
Deep Space Astronauts, Alice, Bob, Charlie and Joe are having dinner in their base station (Deep1), when a 'DeepNote' from their colleague Dot popped up on the DeepHologram in the dining hall. Dot is stationed on Deep2, at a constant 1.66 light years from Deep1. Dot's 10 year service award for Deepspace service is coming up in the not too far future.
Alice, Charlie and Joe are all due to depart soon in order to relieve Dot and other more distant colleagues, for their scheduled returns to Earth. After dinner, the friends decided to surprise Dot by simultaneous arrival, flyby's and messaging at her 10 year service award function. Here is the broad plan that they came up with (overview graphic one the right, with a more readable one copied below).
Bob will stay at Deep1, Alice will use her trusted old 1g continuous acceleration spaceship, thrusting halfway to Deep2, then reversing the thrust for 1g deceleration, in order to come to rest at Dot's station for attending the award function personally. Joe will use the brand new nearinstant boost warpdrive, departing simultaneously with Alice, accelerating momentarily to a constant speed that allows him to fly past Deep2 just when Alice arrives there.
Charlie will use the modern 3g continuous deep space drive to play a relativistic "tortoise and hare" game with Joe  departing somewhat later than the other two and then continuously accelerate at 3g. His departure must be timed so that he will pass Deep2 exactly when the other two arrive there. The three travelers will be monetarily colocate with Dot at the function.
Not to be totally outdone, 'stayathomeBob' will send congrats using the Deephologram system, but also timed so that his message will reach Dot simultaneously with the travelers.
The friends ask Quant1 (Deep1's quantum computer) for a proposal on departure times. Here is what Quant1 came up with.
1. Alice (red curve) is to set off tomorrow at t=T0 on her Deepsynchronized ship clock, accelerating at 1g until t=T0+1.22 yr on her clock and then reverse the thrust. Her maximum speed will be 0.836c relative to Deep1/2 and she will decelerate until she parks at halt at Deep2, at T0+2.44 yr on her clock. It will be T0+3.08 yr Dottime.
2. Joe (gold) is to set off at the same time as Alice, nearinstantly warp to a relative speed of 0.534c and fly past Deep2 at 2.6 yr, also at 3.08 Dot time, as planned. Interestingly, Joe will initially outrun Alice, as she is still accelerating, but somewhere Alice will pass Joe again and then he will catch op again at the moment Alice stops at Deep2.
3. Charlie (the green hare) is to relax and train in the centrifuge for the upcoming 3g ordeal, until T0+1.12 year when he must set off at a constant 3g acceleration, until he flashes past Deep2 at T0+1.94 yr onboard (also at T0+3.08 yr Dot time), having reached the incredible speed of 0.986c and still accelerating.
4. 'Bobthereference' must send his hologrammessage at T0+1.42 yr, exactly 1.66 years before his friends will reach Deep2. This is how long the signal will take to get there, because sadly, instant quantum data transfer still doesn't work.
Note that everyone arrives at Deep2 at the same coordinate time, as can be seen from their identical spacepropertime path lenghts (yes, they are really the same).
Also note that righthand thick purple vertical line is not only Dot's worldline, but represent the event when her friends arrive. This is an important point: in spacepropertime, events are vertical lines, not single points like in a Minkowski diagram. This is the price one pays for the otherwise superb clarity of the uniform scale of space propertime worldlines.
Once this is well digested (?), I will move on to show how spacepropertime couples to gravity through Einstein's equivalence principle and other factors.
J

Re: SpacePropertime Structure and Acceleration II