# relativity and accelerating BBs

#### dseppala

Scenario:
There are three inertial reference frames: F0, F1 and F2. F1 is moving along the x-axis in the negative x direction with respect to F0 with a speed V = sqrt(3)/2*c. F2 is moving along the x-axis in the positive x direction with respect to F0 with a speed V = sqrt(3)/2*c.
In frame F1, there are two BB's at rest positioned along the x-axis 10 meters apart as measured by observers in F1. Per relativity observers in frame F0 measure the separation between the BBs to be 5 meters. At time t0 observers in frame F0 simultaneously start the acceleration of each of the two BBs in F1. They accelerate at a constant rate and continue at this constant rate until they have zero velocity with respect to frame F2.
Using relativity concepts observers in F0 always measure the separation between the two BBs (center to center) to be 5 meters throughout the journey from F1 to F2. Observers in frame F1, observe that one BB started accelerating before the other BB. They observe that during the journey from F1 to F2, the two BB's move closer and closer to each other. Per relativity observers in frame F2 observe that the BBs were initally just under 1.5 meters apart (as measured in frame F2) before the acceleration started and when the acceleration of each BB started their acceleration was in the opposite order so the BBs move further and further apart (as measured by observers in F2) in their journey from F1 to F2 until they are 10 meters apart when they reach F2.
Everyone agrees that is the correct result of relativity. But what is puzzling is what a co-accelerating observer traveling with the BBs measures during the journey from F1 to F2. The acceleration rate has negligible effect on the separation measurements made during the journey just described , but in discussing this scenario with others there seems to be concerns how the acceleration affects the measurements made by the co-accelerating observer traveling along with the BBs on the journey from F1 to F2. So we'll make the acceleration rate very small, say a constant 0.0001 g (as measured in F0) so that it becomes self-evident that the acceleration rate has a negligible effect on the measurements made by the co-accelerating observer.
Now the co-accelerating observer has a 10 meter measuring rod that he generally keeps perpendicular to the x-axis and direction of acceleration. Before the acceleration starts he rotates the rod so that it is aligned with the x-axis. Let's say it takes him 1 second to rotate his measuring rod. He measures that the separation between the BBs is the same length as the 10 meter measuring rod. He then rotates the rod back so that it is once again perpendicular to the x-axis (and thus perpendicular to the direction of motions between the frames in this scenario). Now the acceleration starts. Just after both BBs start accelerating, the co-accelerating observer once again rotates the rod so that it is aligned with the x-axis. He observes that the separation between the two BBs is slightly less than 10 meters. After the measurement he once again returns his measuring rod perpendicular to the x-axis.
Every now and then the co-accelerating observer rotates his measuring rod to align with the x-axis to measure the distance between the BBs. As he travels from F1 to F0, he measures that the separation between the BBs is getting smaller and smaller. When he has zero velocity with respect to F0, he measures that the BBs are only 5 meters apart. But as the co-accelerating observer and the BBs continue their acceleration from F0 to F2, the co-accelerating observer measures that the separation between the BBs is now increasing with each measurement instead of decreasing. When the co-accelerating and BB's have zero velocity with respect to F2 and the acceleration has stopped, the co-accelerated observer measures that the BB's are once again 10 meters apart.
The puzzling question is what caused the BB's to move toward each other during the first half of the journey as measured by the co-accelerating observer and then move away from each other during the last half of the journey? There was no change in the direction of the force on the BB's, the acceleration pattern was identical throughout the journey (although per Einstein, the frames do not agree on which BB started accelerating first), accelerometers next to each BB did not show anything that would cause the BB's to move toward each other during the first part of the journey and then start moving away from each other during the last part of the journey. All the measurements made were simple mechanical measurements.
So how do physicists explain this phenomena? Any insights would be appreciated.

Thanks,
David Seppala
Bastrop TX

#### Woody

Would you post the calculations you use to arrive at these values please.
I have to admit I am too lazy to be sitting down and working it all out for myself,
but if you write it out for me I will try to follow it through.

As you say the relative timings of the events and the relative spacing would depend on the relative motions of the observation frames,
but I am not convinced you have the scenario calculated completely correctly, particularly for the co-accelerating observer.

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1 person

#### dseppala

With V = sqrt(3)/2 * c, that gives a length contraction factor of 2.

The BBs are initially at rest 10 meters apart as measured in frame F1. F0 and F1 have a relative velocity of V = sqrt(3)/2 * c, so frame F0 measures the BBs to be 5 meters apart.

F0 starts the accelerations of the two BBs simultaneously, so throughout the journey, F0 says the BBs remain 5 meters apart (center to center).

When they arrive and come to rest in F2, F0 measures that they are 5 meters apart and due to length contraction they must be 10 meters as measured in F2.

Now the co-accelerating observer has a 10 meter rod that is initially at rest in the perpendicular direction to the motion in this scenario. He makes occasional measurements by aligning the rod in the acceleration direction for a very short time and then puts the rod perpendicular to the acceleration direction until he makes another measurement of the separation between the BBs. The co-accelerating observer when using that 10 meter rod, always makes the same measurements that would be made in the inertial reference frame he has zero relative velocity with respect to when he makes the measurements.

So when the co-accelerating observer has zero relative velocity with respect to F1 he measures the separation between the BBs to be 10 meters just as someone in F1 does.

When the co-accelerating observer has zero relative velocity with respect to F0 he measures the separation between the BBs to be 5 meters just as someone in F0 does.

When the co-accelerating observer has zero relative velocity with respect to F2 he measures the separation between the BBs to be 10 meters just as someone in F2 does.

So during the journey, the co-accelerating observer measures the separation between the BBs to go from 10 meters down to 5 meters, but then as the acceleration continues he measures the separation between the BBs to go from 5 meters back to 10 meters. But the acceleration rate and direction remain the same throughout the journey.

David Seppala
Bastrop TX

#### Woody

I think that the flaw in your description is that the observer in F0 always sees the BBs as being 5m apart.

He will actually see them get further apart until when they pass him they will be 10m apart,
and then they will appear to get closer together again until they are 5m apart again at when they reach F2.

1 person

#### dseppala

Frame F0 always measures the separation (center to center) of the two BBs as 5 meters. When they are at rest in F1 observers in F0 measure them to be 5 meters apart. Then observers in F0 simultaneously start the acceleration of both BBs, so they always remain 5 meters apart as measured in F0 since the acceleration rate is the same and their accelerations were started simultaneously.
David Seppala
Bastrop TX

#### donglebox

Hello friend

My limited experience in numerical computation can only help you think in some directions.

In the experience of solid motion system, people will use an absolute coordinate system to derive other relative coordinate systems. These relative coordinate systems are calculated on the basis of absolute origin, and the transformation between them does not consider the transformation of space-time deformation.

(There may be more macroscopic coordinates such as GPS positioning, but they can still be seen as absolute positions, based on which relative positions can be obtained)

Please make sure that your question applies to the macro scenario of relativity. Their complexity scales are different.

In order to deal with the macro scale, because it is impossible to find an absolute point in macroscopical time and space, I think this is also one of Mr Einstein motivation to set the speed of light to be the scale of Relativity .

I also want to know if I'm wrong with this understanding.Any one， Please point out, thank you.

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#### Woody

The length contraction is a function of relative velocity,
as you have already correctly calculated from F0 to F1 and from F0 to F2

However you have to recognize that there is another frame of reference in your scenario
The frame of reference of the BBs which I will call Frame FA (A for acceleration).

This frame is constantly changing its relative velocity relative to the other frames,
thus the length contraction (as observed from the other Frames) is also continually changing.

Note that the observer traveling with the BBs is also in Frame FA and so will always observe a constant separation.

1 person

#### dseppala

Yes, but why does the co-accelerating observer observe that the two BBs continually move closer and closer to each other and then suddenly they start moving away from each other during the last part of the journey. What caused their motion to suddenly change?

David Seppala
Bastrop TX

#### dseppala

Woody wrote:
"However you have to recognize that there is another frame of reference in your scenario
The frame of reference of the BBs which I will call Frame FA (A for acceleration).

This frame is constantly changing its relative velocity relative to the other frames,
thus the length contraction (as observed from the other Frames) is also continually changing.

Note that the observer traveling with the BBs is also in Frame FA and so will always observe a constant separation."

Woody,
In the scenario I posted, the observer traveling with the BB's does not observer a constant separation between the BBs. He has a measuring rod that is generally kept perpendicular to the acceleration direction. Every now and then he rotates the rod to align it in the direction of acceleration. At the beginning of the journey he measures that the rod is 10 meters in length and the BBs are separated by 10 meters. After the measurement he returns the rod to it perpendicular direction relative to the acceleration direction. When he has zero velocity with respect to F0, he rotates the rod so that it is aligned in the direction of rotation. He finds that the BBs are now only separated by half the length of the 10 meter rod.

David Seppala
Bastrop TX

#### Woody

Oh no he doesn't!

What gives you the idea that the accelerating observer
(who is in the same reference frame as the BBs)
will see any change in their separation?

Observers from other frames will see his rod change length as he swings it from perpendicular to parallel to the direction of acceleration.

But to the observer in FA, it will always look 10m long.

It is only when the observer is in a different inertial frame (is moving at a different velocity) to the object(s) that are being observed that anything "unusual" is observed.