The situation is the same as before, be sure to check out Paradox V1 if you want to be reminded, the same methods will be used as the ones already established. Now we're going to be looking at Apollo out, starting at Destiny not homey which then decelerates with constant negative acceleration (this is unlike what we had before which was just instant acceleration). So the time at which we call t_turningpoint, the ship attained velocity 0 and is returning to destiny.
If we look at the event where the velocity is 0, we see that the time is the same for the astronaut and at the planet, why must this be the case? Well they are moving with the same velocity so they have the same time, no time dilation occurs at that one instant. Or we can say that she is in the frame of reference of the planet.
If we plot time against time (in two different frame of reference we get the following
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My laptop decided it's enough plotting today, we're doing this shit by hand!! At 202 years Lisa stars to negatively accelerate and what happens is that she will observe that time goes much faster on homey's clock than before she began accelerating. We found that Lisa took 296 years to reach the turning point, using spacetime intervals we found that. Last time we saw that it took 4 years but here it took 296 years so all in ask: 292on homey's clocks. Using symmetry we know that the time going is the same as returning meaning that she must observe that it took 292 years going back, she will end up with the same speed that she started with before beginning to accelerate. In total it took her 588 years to Destiny and 592 back to homey
To the important question, how much did Lisa age?
Let's talk about how we did it first though:
We will assume an inifinite amount number of space ship elevators going back towards Destiny and beyond (to Beyond), all with velocities that vary from 0 to v_0, and Lisa stays in one elevator for a small time then jumps on to the enxt one going a tad bit faster, and so on and so forth until she reached the one with v_0, We rewrote time dilation in terms of acceleration times time instead of the usual velocity which helped us set up a nicer solvable integral, then we pull a physics move (mathematicians be mad) and integrate both sides, obtaining the total change in time from t = 0 to t = v_0 /g, we find then that it took 74.5 years during the acceleration fase.
Okay putting all of this together, we get that in
the spaceship frame Lisa used 28.5 years from Homey to Destiny, 74.5 years under the acceleration phase to turning point and 74.5 years back to destin, then 28.5 back to homey, we obtain 206 years in total.
Observer chilling at homey frame Remember that this fella just read the clock on homey and he read that it took 592 years all in all, 4 years to destiny, 292 years from destiny to turning point and 292 to destiny again then 4 years from destiny back to homey.
Planet frame It took straight up 296 years the whole way to turning point and 296 years back, giving us 592
Can you believe it? This is perfect, let's wrap it up.
What makes this seem like a paradox is that in the beginning they were not agreeing on their respective time measurements, but now we see that they do. In the acceleration phase Lisa changes frames of Reference infinitely many times (integral time), and for each change in speed there is a small time dilation associated, we sum them all up to find that it takes 149 years in her own system, or 206 in total. The same two events had a time interval of 592 years on the planet frame. Ah feels good to be done with this shit, man I hate relativity. But yes, Lisa's sister is 592 years older (steroids) and Lisa is 206 years older. No paradoxes in physics babyyyy