EDIT (to clarify my question): I think some of the answers here are accounting for light travel time and telling me what I'd actually see on Earth's clock, so I've edited my first paragraph to clarify. I don't think this clarification changes the meaning of this question, but it might.
Summary: if I accelerate to $0.8 c$ in 1 second, how much time passes for observers in my starting inertial reference frame?
This seems like a simple question that has probably been answered, but I couldn't find a simple answer for what appears to be a simple question:
- I start 8 light years from Earth, at rest with respect to Earth, and observe Earth's time is t=0. Of course, technically, I'm seeing Earth the way it was 8 years ago (t=-8), but I know I'm 8 light years away from Earth, so automatically add 8 years to the time I see.
I make this assumption throughout the question. In other words, when I say "Earth's clock time", I mean: "the time I'm seeing on Earth's clock right now plus my distance from Earth in light travel time".
I believe this is the norm in relativity questions, but could be wrong about that.
Keeping an eye on Earth's clock, I accelerate to $0.8 c$ in 1 second. Because I'm accelerating, I know Earth's clock will go faster than mine. The question is: how much faster, and where will it end up after I've finished my one second of acceleration to $0.8 c$?
At $0.8 c$ the distance to Earth is now 4.8 light years (minus the little bit I traveled during acceleration). Earth's clock now runs slower than mine by time dilation. So, when 6 of my years have passed, fewer than 6 years have passed on Earth's clock.
As I get close to Earth, I "decelerate" to the Earth's reference frame so that I will be at rest when I actually arrive at Earth. Of course, deceleration is just acceleration in a different direction, so, once again, Earth's clocks run faster than mine.
And, once again, the question is: in that 1 second of deceleration, how much time elapsed on Earth's clocks?
What vexes me about this problem:
In the 6 years I was traveling at $0.8 c$, Earth's clocks ticked off only 3.6 years by time dilation.
By the time I arrive at Earth, Earth's clocks must have ticked off 10 years, since they say me traveling at 0.8c for (most of the) 8 light years.
The only way I can reconcile these numbers (10 years minus 3.6 years, or 6.4 years) is that my 1 second of acceleration and deceleration each took 3.2 Earth years (about 10^8 seconds).
This seems high, and I can't get the numbers/formulas to yield this, but...
On the other hand, it seems somewhat reasonable that the amount of time that passes depends only on my final velocity ($0.8 c$) and not how fast I reached that velocity.
Note that I don't think there's a simultaneity issue here, since I start and end in Earth's reference frame.