# Calculating Time dilation in Accelerated systems

1. Jan 1, 2010

### Austin0

From reading in this forum I have gotten an idea regarding the calculation of dilation between an accelerating frame and an inertial frame.
It seems there are two methods I have heard of:
A) An evaluation based on the sum of instantaneous or infintesimally short interval relative velocity derivations.

B) A direct line integration of the curved path of the accelerated frame in Minkowski space.

AS I understand it, the derivative method produces a result that agrees with empirical testing I.e. No dilation beyond what is expcted to result from relative velocity.
Whereas the integration method produces a greater quantitative result than either the derivitive method or the actual test results.
I.e. AN additional dilation factor resulting from acceleration per se. SO it does not conform to the real world.

Is this view essentially correct or am I just misunderstanding and both methods return the exact same quantitative results for any given situation.
ANy clarification appreciated as the practical calculus is beyond me. Thanks

2. Jan 1, 2010

### Al68

Unless I misunderstand, A) calculates the rate of a different non-local clock relative to an accelerated clock during the intervals, while B) calculates the total elapsed proper time of an accelerated clock between events, so it's like comparing raw apples to cooked oranges.

That being said, even though they are calculating different things, they should both match empirical results if done correctly.

Last edited by a moderator: Jan 1, 2010
3. Jan 1, 2010

### Austin0

Originally Posted by Austin0
From reading in this forum I have gotten an idea regarding the calculation of dilation between an accelerating frame and an inertial frame.
It seems there are two methods I have heard of:
A) An evaluation based on the sum of instantaneous or infintesimally short interval relative velocity derivations.

B) A direct line integration of the curved path of the accelerated frame in Minkowski space.

Yes I think you misunderstood. As far as I know the derivitive method is a cumulative sum arriving at total elapsed time. I have encountered this method both in this forum and the analysis of various actual tests.
I agree that from my limited knowledge of calculus that they both should produce the same result that is why I am unsure of the idea I have gained through various threads in the forum [which i now have no idea the location of] that this was in fact not the case. That the additional distance due to the curvature of the world line and the resulting calculated overall dilation is not in agreement with tests. That the method summing instantaeous direct relative velocity vaues does agree with tests.
So they should be calculating exactly the same thing.
In any case Thanks for your responce .

4. Jan 1, 2010

### Mentz114

Austin0,

the two methods you cite seem to me to be the same thing, because integration is the summing of infinitesimal elements.

In practice, if an observer watches the other clock through a telescope for one minute by his own clock, and notes the elapsed time on the other clock, then this observer is comparing a 1-minute section of his worldline with a piece of the other clocks worldline. Both clocks are just measuring off ( integrating) the proper lengths of their own worldlines.

5. Jan 1, 2010

### Staff: Mentor

I agree, integration is a sum of infinitesimal segments.

6. Jan 1, 2010

### Al68

Sure, the results of A) can be integrated between events to get elapsed time, but I assumed you were referring to using the accelerated reference frame for method A), which means the clock at its origin would have a velocity of zero in that frame. So there's no reason to use A) in the accelerated frame to calculate the elapsed time of that clock, since it would end up being T*(1)=T. The only reason to use A) in the accelerated frame is to calculate the rate of a different clock than the one at its origin.

In other words, there is no "time dilation in an accelerated system" for a clock stationary at the origin of that system. The time dilation factor is equal to one for that clock, since it's used as reference. It's only other clocks that are "time dilated" in the accelerated frame.

Last edited by a moderator: Jan 1, 2010
7. Jan 2, 2010

### Austin0

Totally agreed. This exactly why I was surprised when I encountered the assertion that the line intergration method did not produce correct results. That is why I am trying to get clarificatin of this question now.I apologize for not having references to the threads where I have encountered this idea. But it was more than once.
The geometric integration was presented as evidence that there should be dilation attributable to the increased length of the worldline due to the deviation from a strictly linear path and this being countered by the empirical evidence and the counter that it didnt agree with the instaneous summing of relative velocities.

Thanks

8. Jan 2, 2010

### Austin0

The setup is simple . Working with a diagram of a course of acceleration and an inertial frame with the inertial frame being the rest system you can then apply either of these methods to calculate the relative proper time interval of the accelerated system conpared to the elapsed time of the inertial system which is directly read from the diagram. AS you point out the system that is the rest system has no dilation.
As I understand it :
A) is purely mathematical. Taking an increasing gamma factor based on instantaneous relative velocity applied to short intervals of the inertial timeline and adding up the total to derive the interval of the accelerated system.
B) Is geometric. Adding up segments of the accelerated worldline to get a geometric length and then converting this to a time interval . And then comparing to the inertial elapsed interval.
Since the logic that a curved line must have a greater length compared to a straight line between the same points seems incontrovertable, I am in agreement with all of you and think it should be accurate.
But besides what I have read in this forum there is the fact that it appears acceleration does NOT directly induce dilation so I am still left i a state of uncertainty and hope for a definitive answer to this question. Thanks

Last edited: Jan 2, 2010
9. Jan 2, 2010

### Al68

Sorry, I did misinterpret your initial post. I assumed that by "accelerated system" you meant accelerated reference frame.

It looks to me that if B) is done correctly, it would just be a geometric representation of A), otherwise nobody would use B).

I've never seen a SR/GR interpretation that claims that acceleration directly causes time dilation. To the contrary, the clock hypothesis is assumed, so that the operation of a clock must be unaffected (directly) by acceleration to be a valid clock.