But this is not the end of the story. Infinity is never the correct answer in physics. Infinities in physics must either cancel out or approach a real number limit. The real number limit in this case is 2. Alice choosing +c away from Bob at the turnaround means she will get to 2 yrs older than Bob during the imbalance period and stay only 2 yrs older so long as she stays at c. There are 2 ways to work out this answer.
1. Empirically: If you do the analyses of Alice choosing .8c, 15/17 c, 40/41 c and .99969 c away from Bob, they intersect their half speed velocity lines (.5c, .6c, .8c and 40/41c) from Bob = 8 to Alice = 8.5, 8.8, 9.4, 9.92. It looks like the closer to +c she flies away from Bob, the closer her age difference approaches +2 (not infinity).
2. Mathematically: It looks like the ratio of full speed to half speed velocity lines has something to do with this limit. Yes Y approaches infinity for both these velocity lines but their intersection is between a line that has Y infinity and another which has half Y infinity. This suggests the answer is 2 but I have not yet figured out how to cancel out the infinities using fancy math tricks.
Note: This final answer and analysis has been edited from the answer I gave last time. Relativity does not allow calculation of age difference in scenarios where Alice does not return to Bob (because of it's assumption that perspective time is real, (not illusion due to perspective), and defines the present (which is subjective and individual to everyone). I believe causal time is the true reality from which perspective reality follows. Causal time is not absolute time, it is also relative.
Here is the STD of Alice making velocity choices beyond +.6c.
The intersection points for .8c, 15/17c and 40/41c move farther and farther apart but so do the units on those velocity lines due to the increase in Y so the resultant trend in age difference doesn't go to infinity as Alice's velocity approaches c, it tends to the number 2.
Here's the math for .99969 c which is not shown on the STD.
Y = 40.506
(x-3)/t = .9996952148735142c (3280/3281c)
x/(t-3) =40/41c (half speed of .99969c)
x = ((3 + 3(.99969))40/41)/(.99969 - 40/41) = 243
t = (x -3)/.99969 = 240
t' = 240/Y = 5.92 add 4 (Alice's age at turning point) = 9.92 (Alice's age when Bob gets the news) and subtract 8 (Bob's age) and the age difference is 1.92 yrs Alice is older than Bob.
Just be thankful the math doesn't involve anything beyond arithmetic and basic high school algebra.
Edited by ralfcis, 13 March 2019 - 06:06 PM.