A matter of time dilation: how much time has actually passed?

[bbbl67]

TL;DR Summary

How much time differential will two relativistic ships actually see between each other if they accelerate at different rates?

Two relativistic spaceships, A & B, are launched simultaneously from Earth towards Kepler 22b, 640 LY away. Ship A is a lighter ship, has no life support as all it carries is some powered-down androids, and it can accelerate at a constant 3g. Ship B is a more massive ship, it carries human crew, and it accelerates at only a constant 1g, to make it more comfortable for the crew. Both ships accelerate constantly for half the trip, and then flip around and decelerate at the exact same rate for the second half.

Ship A takes 640.6 years to reach Kepler 22b, as measured from Earth; on board, only 4.904 years have passed; it reached a maximum speed of 0.9999994919c at the half-way point before starting the deceleration phase. Ship B takes 641.9 years to reach it, as measured from Earth; on board, 12.586 years have passed; it reached a maximum speed of 0.9999954455c before deceleration.

So measured by each ship's onboard clock, there is a time differential of roughly 7.7 years between the arrival times of ships A vs. B. But measured from clocks on Earth, the time differential between each one's arrival at Kepler 22b is only 1.3 years. Assuming that the local time dilation between Earth and Kepler 22b are roughly equal, will the androids see the humans arriving on Kepler 22b: 7.7 years later, or 1.3 years later?

 

[DaveC426913]

Just extracting the data from the narrative for clarity:

Ship A:
a: 3g
v .9999994919c.
t(earth): 640.6 years
t(subjective), 4.904 years

Ship B:
a: 1g
v: .9999954455c
t(earth): 641.9 years
t(subjective): 12.586 years

time discrepancy (subjective): 7.7 years
time discrepancy (earth): 1.3 years


Did I miss anything?
I have not independently verified any of your numbers yet, just transcribed it.


One thing seems odd off the top of my head: Ship B is restricted to only 1g for the entire trip, yet manages to arrive a mere 1.3 years later than ship A?

 

[bbbl67]

Did I miss anything?
I have not independently verified any of your numbers yet, just transcribed it.

No, your summary seems right.

If you want to verify my numbers, I used to following online calculator:
https://www.omnicalculator.com/physics/space-travel

One thing seems odd off the top of my head: Ship B is restricted to only 1g for the entire trip, yet manages to arrive a mere 1.3 years later than ship A?

Yes, that's because the 1g and 3g start to converge once you get to relativistic speeds, since you can't exceed the speed of light.

 

[PeroK]

One thing seems odd off the top of my head: Ship B is restricted to only 1g for the entire trip, yet manages to arrive a mere 1.3 years later than ship A?

After about a year of 1g acceleration, ship A will be close to $c$ relative to Earth. So, ship A takes only about 2 years more than light to get to Kepler 22B (one extra year for the deceleration phase as well). This is largely independent of how long the journey is. Ship B takes only 0.6 years more than light, as it travels at close to $c$ for almost all the journey.

will the androids see the humans arriving on Kepler 22b: 7.7 years later, or 1.3 years later?

1.3 years.

 

[Orodruin]

TL;DR Summary: How much time differential will two relativistic ships actually see between each other if they accelerate at different rates?

Assuming that the local time dilation between Earth and Kepler 22b are roughly equal, will the androids see the humans arriving on Kepler 22b: 7.7 years later, or 1.3 years later?

You are confusing times here. 7.7 years is the difference between the proper travel times of the humans and androids. 1.3 years is the time passing at the destination between the androids arriving and the humans arriving. Therefore, the androids will have aged 7.7-1.3 = 6.4 years less than the humans when they meet again as the humans arrive.

 

[Ibix]

Without checking the numbers, the timeline using Earth clocks is:

• T=0 - departure
• T=640.6 - ship 1 arrives
  • Ship 1 clocks read 4.9
• T=641.9 - ship 2 arrives (1.3 Earth years after Ship 1)
  • Ship 1 clocks read 4.9 + 1.3 = 6.2 (because its clocks tick at the same rate as Earth clocks now itcs at rest at its destination)
  • Ship 2 clocks read 12.6 (6.4 years more than Ship 1, consistent with @Orodruin's calculation)

So the first ship only has 1.3 years' head start in exchange for even more insane fuel costs than the 1g trip. Just send then earlier...