Can Time Dilation be Determined in a Constantly Accelerating Frame?

[JesseM]

This is an acceptable change to the problem but I don't see why it is necessary.

It isn't "necessary", but as I said a few times, it makes the math simpler. If you wanted a period of finite acceleration, you'd have to first find the velocity as a function of time in some inertial frame during the acceleration, then integrate $$\int^{t_1}_{t_0} \sqrt{1 - v(t)^2/c^2} \, dt$$ in that frame to find out how much time will elapse on the ship's clocks during the acceleration. If you specify constant acceleration from the point of view of the ship, then the rate of acceleration will not be constant from the point of view of an inertial reference frame, so v(t) won't just be a*t.

t=101 as for the case you outlined above. The point has always been that S1, S1', and S2' decelerate from the S' frame to the S frame simultaneously as seen by the S frame, that is the S2 mother ship. All I've asked for are the time for S1' before it instantaneously decelerates and the time for S2' after it instantaneously decelerates. Do not forget the details of the problem when you make these calculations.

Sounds good. Assuming all three decelerate at the same moment in frame S, the clock of S1 will read 100 + (0.4359)(1) = 100.4359 seconds, while S1' and S2' will read (101)*(0.4359) = 44.025 seconds. In frame S', the event of S1 decelerating (I'll keep calling it 'decelerating' even though in this frame the velocity increases) happens at a different time than the event of S1' and S2' decelerating; S1 decelerates at time t'=229.852 in this frame (found by doing a Lorentz transform on the coordinates x=0.9, t=101 in the S-frame), while S1' and S2' decelerate 185.827 seconds earlier at t'=44.025 (found by doing a Lorentz transform on the coordinates x=90.9, t=101 in the S frame). Since they run at 0.4359 the normal rate after decelerating in this frame, they will read 44.025 + (0.4359)*(185.827) = 125.025 at the moment S1 decelerates, in the S' frame.

On the other hand, you might want to specify that all three decelerate at the same moment in frame S' instead. In this case, it's still true that the clock of S1 will read 100.4359 seconds, but now S1' and S2' will read 229.852 seconds at the moment they decelerate (again, found by doing a Lorentz transform on x=0.9, t=101 in the S frame). Now in the S frame these events no longer happen at the same moment--instead, at the moment S1 decelerates, S1' and S2' have not yet decelerated, and they only read (0.4359)*(101) = 44.025 seconds at that moment.

 

[Aer]

Assuming all three decelerate at the same moment in frame S, the clock of S1 will read 100 + (0.4359)(1) = 100.4359 seconds, while S1' and S2' will read (101)*(0.4359) = 44.025 seconds. In frame S', the event of S1 decelerating (I'll keep calling it 'decelerating' even though in this frame the velocity increases) happens at a different time than the event of S1' and S2' decelerating; S1 decelerates at time t'=229.852 in this frame (found by doing a Lorentz transform on the coordinates x=0.9, t=101 in the S-frame), while S1' and S2' decelerate 185.827 seconds earlier at t'=44.025 (found by doing a Lorentz transform on the coordinates x=90.9, t=101 in the S frame). Since they run at 0.4359 the normal rate after decelerating in this frame, they will read 44.025 + (0.4359)*(185.827) = 125.025 at the moment S1 decelerates, in the S' frame.

So, S1' and S2' decelerate to the S frame before S1 enters the S' frame according to an observer stationary in the S' frame. Meanwhile, while S1 is in the S frame before accelerating to the S' frame, an observer in the S frame will say that S1' and S2' are still in the S' frame. So no matter what, over this infintesimally small instant of acceleration, S1 managed to escape being in the same frame as S1' and S2' even though S1' and S2' didn't decelerate to the S' frame at the same time according to either an observer in the frame S or S' as S1 accelerated to the S' frame. I am not saying this is a contradiction per say, I am only asking if this is an accurate assessment of the situation or if it needs to be revised.

On the other hand, you might want to specify that all three decelerate at the same moment in frame S' instead. In this case, it's still true that the clock of S1 will read 100.4359 seconds, but now S1' and S2' will read 229.852 seconds at the moment they decelerate (again, found by doing a Lorentz transform on x=0.9, t=101 in the S frame). Now in the S frame these events no longer happen at the same moment--instead, at the moment S1 decelerates, S1' and S2' have not yet decelerated, and they only read (0.4359)*(101) = 44.025 seconds at that moment.

Let's not consider two different scenarios.

 

[JesseM]

So, S1' and S2' decelerate to the S frame before S1 enters the S' frame according to an observer stationary in the S' frame.

Yes.

Meanwhile, while S1 is in the S frame before accelerating to the S' frame, an observer in the S frame will say that S1' and S2' are still in the S' frame. So no matter what, over this infintesimally small instant of acceleration, S1 managed to escape being in the same frame as S1' and S2' even though S1' and S2' didn't decelerate to the S' frame at the same time according to either an observer in the frame S or S' as S1 accelerated to the S' frame.

It's a little ambiguous to talk about what happens over an infinitesimally small period of time--the velocity is really undefined at the exact moment of an instantaneous acceleration, and so the instantaneous inertial rest frame at that moment is also undefined. If you made the acceleration non-instantaneous, then S1 would have a different instantaneous rest frame at each moment during the acceleration, and each of these instantaneous rest frames would have a velocity midway between S and S'. Note that this would be just as true in Newtonian mechanics as it is in relativity.

 

[pervect]

This is an irresponsible post on your part.

Sorry you feel that way. I'm about 5-10 posts behind, so perhaps you and Jesse have already addressed the things that were bothering me. I just don't know, because I don't have time to follow this thread in the amount of detail that it would take me to figure out whether or not you two had already covered my misgivings or not.

I thought I'd just let everyone involved know that I had to "abandon ship" due to time constraints, rather than just walk away from the thread without telling anyone.

 

[Aer]

Yes. It's a little ambiguous to talk about what happens over an infinitesimally small period of time--the velocity is really undefined at the exact moment of an instantaneous acceleration, and so the instantaneous inertial rest frame at that moment is also undefined. If you made the acceleration non-instantaneous, then S1 would have a different instantaneous rest frame at each moment during the acceleration, and each of these instantaneous rest frames would have a velocity midway between S and S'. Note that this would be just as true in Newtonian mechanics as it is in relativity.

Very well, my purpose to this thread was to gather a better understanding of all the implications of special relativity as I've merely only had an introductory lecture on it many years ago. I still have concerns about the notions of space contraction as described by the lorentz transformations as I alluded to in another thread as well as the issue of simultaneity as illustrated in this thread. These concerns originated from an idea (I would hesitate to refer to it as any type of theory at this point) on the fundamental nature of matter which would be a better topic for the Strings, Branes, & LQG forum. This idea would explain a lot of the experiments on special relativity but I noticed right away it would not allow for some of the ideas purported by special relativity. Anyway, thank you for your time.

Sorry you feel that way. I'm about 5-10 posts behind, so perhaps you and Jesse have already addressed the things that were bothering me. I just don't know, because I don't have time to follow this thread in the amount of detail that it would take me to figure out whether or not you two had already covered my misgivings or not.

I thought I'd just let everyone involved know that I had to "abandon ship" due to time constraints, rather than just walk away from the thread without telling anyone.

I can understand time restraining anyone from delving deeply into a certain topic, there is no need for you to apologize for this.