# Time dilation and expansion in accelerated motion

1. Jan 11, 2007

### MeJennifer

Please carefully consider the following two experiments:

Experiment 1
In flat space-time, two completely identical small test probes with built-in rockets and computerized navigation equipment separated by an initial distance l accelerate with a constant proper acceleration a for a proper time interval t. Each probe records the proper duration of the acceleration.
An observer fetches both records and compares the durations as was recorded.

Experiment 2
In flat space-time, one end of a metal rod of a length l is accelerated away from the direction of the rod, in other words the other end is trailing, with a constant proper acceleration a for a proper time interval t. Two completely identical small test probes with a built-in synchronized ideal clock, ideal accelerometer and computerized navigation equipment were placed at each end of this rod. However the probe in the trailing rod only records, its rocket engine is disabled. Each probe is individually programmed to record both the time when the acceleration started, the proper duration of the acceleration and the proper acceleration profile.
An observer fetches both records and compares the start times as well as the proper durations and the proper acceleration profiles as was recorded.

It seems that there is no problem with Experiment 1, each record will show an identical duration and each probe has recorded the same path curvature, one of constant proper acceleration.

My question is what is the difference between the recorded proper times (if any) between the two probes and what is the proper acceleration profile for the probe on the trailing end of the rod.

Last edited: Jan 11, 2007
2. Jan 11, 2007

### FunkyDwarf

Im not sure i understand the senario. You say that one end is accelerated while one trails, surely the entire rod experiences the exact same acceleration? assuming no compression etc (perfect ridgid body)

3. Jan 11, 2007

### Jorrie

Hi MeJennifer,

I understand your experiments, but want to know about the rigidity of your rod - is it Born rigid?

If it is, the 2 experiments are identical to the Bell spaceship paradox and the problem of Born rigid motion. Both have been treated extensively in the literature.

If not, the only difference to the above should be the fact that the trailing end of the rod will start accelerating a time t = l/v after the front end, where v is the speed of sound in the rod. The same will then happen in reverse when acceleration stops.

Regards, Jorrie

4. Jan 11, 2007

### Jorrie

Rigid rod

In a hypothetical perfectly rigid rod scenario, the trailing edge of the rod always has higher proper acceleration than the leading edge, irrespective of which end of the rod has the propulsion.

Look at the latter part of "[URL [Broken]

Last edited by a moderator: May 2, 2017
5. Jan 11, 2007

### my_wan

This does appear to be a version of Bell's Spaceship Paradox. I like the way this site explains it.
http://math.ucr.edu/home/baez/physics/Relativity/SR/spaceship_puzzle.html [Broken]
The main point is that different observers may not agree on wether the two ends of the rods actually began accelerating at the same time. Every paradox in relativity is an application or misapplication of simultaneity.

Last edited by a moderator: May 2, 2017
6. Jan 11, 2007

### Jorrie

The URL you quoted does not actually explain the Born rigid motion, which is more like the experiment (no. 2) that MeJennifer had in mind, I think.

Jorrie

Last edited by a moderator: May 2, 2017
7. Jan 11, 2007

### pervect

Staff Emeritus
There is a space-time diagram for this exact case at http://en.wikipedia.org/wiki/Image:Bell_observers_experiment2.png

As you can see from the diagram, the line segments AB and A'B' will both be horizontal, so events A and B will be simultaneous in the launching frame S, which is the frame used to draw the diagram.

Similarly, when the acceleration ends at events A' and B', the end of acceleration events will be simultaneous in frame S, the "launching frame".

As you should also be able to see from the diagram, events A' and B' will NOT be simultaneous in frame S', an inertial frame co-moving with the final velocity of the two probes.

By defintion, events A' and B' will have the same encoded "proper time".

The event simultaneous with A' on the worldline of the other probe in frame S' (a frame comoving with the probes after they have accelerated) is shown on the diagram - it is event B''.

For more details see for instance the wiki article on the "relativity of simultaneity". Pay special attention to the process of how the "line of simultaneity" is constructed, i.e the dashed line in

http://en.wikipedia.org/wiki/Image:Relativity_of_simultaneity.png

The "relativity of simultaneity" is really a key point here.

http://en.wikipedia.org/wiki/Relativity_of_simultaneity

There is insufficient information to solve this problem as stated - one would need detailed information about the mechanical properties of the rod and how it stretches. If it is intended that the rod be Born-rigid, for example, this needs to be specified (and unless the concept is understood, there may be some confusion about what the specification means, exactly).

Last edited: Jan 11, 2007
8. Jan 11, 2007

### yogi

Jenniferme - You might take a look at Taylor and Wheeler pages 117 and 118 second edition where the problem is labeled "Paradox of the identically accelerated twins." Since "Spacetime Physics" is essentially a text on SR, the issues are resolved by treating the accelerated motion as a series of impulses ...the same result of course as that depicted by hurkyl in the previous thread that raised these same questions

9. Jan 12, 2007

### bernhard.rothenstein

time interval measurement by accelerating observers

Please have a critical look at
arxiv physics/0610226\
arxiv physics/0600049
arxiv physics 0607288

10. Jan 30, 2007

### Boustrophedon

Experiments 1 & 2 yield identical results in special relativity

I would like to point out that almost all the replies so far given to the question posed by Mejennifer are based on Lorentz's pre-1905 theory that postulates actual physical contraction of solid lengths in the direction of motion. It should be remembered that Einstein's SR is a fundamentally different theory even though it leads to the same Lorentz transformations.

Einstein's theory involves a purely kinematical approach involving no physical "shrinkage" but achieving contracted measurements by means of the relativity of simultaneity. That is to say, the shift in simultaneity causes the front end to be measured first with respect to the rear end a moment later, resulting in a reduced measurement.

What this means is that a rod initially at rest with respect to an observer does not, in SR, change its length with respect to that same observer, as it is accelerated to some fraction of c. What happens is that the length of the rod defined by another observer moving with it will appear to get longer with respect to the "stationary" observer who sees the rear end marked increasingly before the front end, as the moving observer's simultaneity shifts.

Einstein's 1905 paper only concludes that a length defined in K' appears shorter in K by the Lorentz factor, and vice versa, where K' and K are in relative uniform motion. It does not say nor suggest that a body would change its physical length during acceleration.

The idea of a rod "shrinking" as it accelerates is an unfortunate anachronism - a "hang-over" from Lorentz's earlier theory that still lingers on a century later and even finds its way into textbooks now and then.

Thus to get back to the question of the two experiments, the actual prediction of special relativity ( not Lorentz's ) theory is that they both give identical results.

11. Jan 30, 2007

### nakurusil

You missed pervect's answer. You need to specify a lot more in order to get the two problems correctly stated. Pervect completed the corrct statement of the first problem and gave the appropriate solution.The wiki page on the Bell paradox solves the problem but you will need to read on Born rigidity quite a bit before you even look at the problem.The second problem needs a lot more work in terms of stating it correctly and completely before an answer can be attempted.

Last edited: Jan 30, 2007
12. Jan 30, 2007

### Jorrie

Missing the point

Sorry, but I think you are missing the point of this issue: it is not about physical contraction.

Mejennifer posed a quite valid problem (given a few more specifications, as Pervect has pointed out), of how clocks behave and how acceleration is measured in differently accelerating frames. This is not a trivial problem, even in a properly specified "Born rigid" scenario.

I'm leaving it to the advisors to settle!

Jorrie

13. Jan 31, 2007

### Boustrophedon

Very much the point !

On the contrary, the fact that Einstein's SR did away with the need for Lorentz's artificial contraction postulate and showed that relative simultaneity led to differently moving observers measuring different lengths for the same object means that the two separate test probes (exp 1) behave identically to the rod with test probe at each end (exp 2). It is that simple !

The analogy with Bell's spaceships/string problem is accurate but Bell's wrong conclusion of string-breaking is based on the argument that "the spaceships must remain at constant distance but the string "needs" (or is "trying to") Lorentz contract". This kind of contraction has no place in SR, where the string and the spaceship distance (or rod and test probes) represent exactly the same "moving length".

It is easy to prove that actual shrinkage of either string or rod is incompatible with SR by considering a plurality of observers, each accelerating up to a different velocity and each measuring a different length for the same object which remains "at rest". It is obviously impossible that any physical shrinkage could make the object be at once any number of different lengths.

MeJennifer's problem is clearly concerned with any relativistic difference in behaviour between a solid length (rod) and an equal space between similarly moving objects. It is a standard premise of such thought experiments that 'ordinary' inertial effects are neglected as they can always be rendered negligable by sufficiently gentle acceleration.

14. Jan 31, 2007

### Jorrie

Not Contraction Question

I still think MeJennifer's original problem statement has nothing to do with Lorentz contraction. She asked how the recorded acceleration time of a front and rear probe would differ and how their acceleration profiles would differ, if at all.

Your apparent position is that in both her experiments the recorded results (acceleration time and acceleration profiles) of the front and the rear probes would be identical. This is certainly not the accepted mainstream position, which has been explained over and over and...

Jorrie

15. Jan 31, 2007

### Boustrophedon

It has nothing to do with Lorentz contraction in the sense that if consideration of physical shrinkage as proposed by Fitzgerald/Lorentz is excluded, then the correct answer according to SR is obtained, i.e. that exp.'s
1 & 2 give identical results. All the arguments that claim differential acceleration along the rod are based on, and derived from, just such inappropriate assumptions of a priori physical contraction.

16. Jan 31, 2007

### Demystifier

Check out
http://arxiv.org/abs/physics/9810017
especially Eq. (26) and its generalization (27).

17. Feb 1, 2007

### Boustrophedon

Yes, I went through the Nicolic paper a year ago and it should be obvious that he assumes implicitly from the start that the rod undergoes actual physical shrinkage of precisely the type postulated by Fitzgerald & Lorentz prior to SR.
It is also further incorrect in claiming a difference between "pushing" and "pulling" even if we allow him to use Lorentz theory: his equations for each case are actually the same equation that just looks different due to the shift in co-ordinates by a distance L.

18. Feb 1, 2007

### Jorrie

Huh?

Where do you read this in the cited paper? I read just before eq. 1:

"We assume that the accelerated rod is rigid, which means that an observer located on the rod does not observe any change of the rodâ€™s length. (In Section 6 we discuss the validity of such an assumption.)"

I also do not understand your problem with pushed and pulled rods. Nowhere is Nikolic referring to proper lengths, except for non-rigid rods:

"In this section we give a qualitative discussion of how the non-rigidity of realistic rods alters our analysis and find conditions under which our analysis is still valid, at least approximately. First, it is clear that, in general, the proper length of a uniformly accelerated rod will not be equal to the proper length of the same rod when it is not accelerated. For example, we expect that a pushed rod will be contracted, while a pulled rod will be elongated."

The other instances did not refer to proper lengths, but rather to observed lengths in some inertial reference frame, which during acceleration, may be different for rigid pushed and pulled rods.

Jorrie

19. Feb 1, 2007

### Boustrophedon

I said implicitly, and the quote you include is perfectly consistent with what I said - of course the co-mover doesn't observe any change in length.

Neither did I refer to proper lengths, so I don't know why you're banging on about them. I merely point out that the equations he presents to distinguish the two cases (pushing & pulling) are just the same equation with a shift in co-ordinates by L. His argument for a difference between the two is empty.

20. Feb 1, 2007

### Jorrie

I still do not understand what point you're making - do you say that an unaccelerated observer will observe (i.e., properly measure the coordinates of the two ends simultaneously in his frame) the length of a pushed and pulled (rigid) rod as identical while the acceleration lasts?

Jorrie