Undergrad Relativistic Rocket: Understanding Behavior & Speed Limit

[exponent137]

A rocket flies past the me with velocity $0,866$ c, therefore $\gamma=2$. Its length in rest is 10 m. When I am parallel with the last part of the rocket, the rocket stops immediately. The last part stays parralel with me, but the beginning of the rocket jumps for factor 2, therefore jumps for 5 meters immediately. This implies that deacceleration should have some limit that it prevent velocity larger or equal than c. But I never heard for such connection between acceleration and speed limit.

One possible objection is that a relativistic rocket does not behave as a rigid body; Lorentz equations do not prevents that all the rockets stops immediately, but except of this $v>c$. Maybe this seems the most logical answer to this paradox.

Where I am wrong, or what a correct answer to this is?

 

[Nugatory]

One possible objection is that a relativistic rocket does not behave as a rigid body; Lorentz equations do not prevents that all the rockets stops immediately, but except of this $v\gt c$. Maybe this seems the most logical answer to this paradox.

That is indeed the answer. This situation is closely related to https://www.physicsforums.com/insights/can-i-send-a-signal-faster-than-light-by-pushing-a-rigid-rod/

 

[exponent137]

But, something is even unclear to me:
Let us say that one point flies parralel to the beginning of the rocket. When this point stops immediately, (when the end of the rocket flies close to me) it is immediately transferred 5 m ahead. But a rigid body is not necessary here, let us say that a rocket flies forward without stopping.

My possible answer is that even one point cannot be stopped immediately, not only a rigid body?

 

[Ibix]

When this point stops immediately, (when the end of the rocket flies close to me) it is immediately transferred 5 m ahead

No it's not. It stops where it is. If you stop the rocket immediately by using multiple powerful engines mounted all along the side of the rocket then you'll get a 5m long rocket. Which will react exactly as you'd expect a rocket compressed into half its own length to react - crumple and probably explode.

 

[exponent137]

No it's not. It stops where it is. If you stop the rocket immediately by using multiple powerful engines mounted all along the side of the rocket then you'll get a 5m long rocket. Which will react exactly as you'd expect a rocket compressed into half its own length to react - crumple and probably explode.

Intuitively, it seems the most logical answer to me. Probably we are in contradiction if your answer is refuted? So thus we ignored rigid body rule in this problem?

 

[Orodruin]

When I am parallel with the last part of the rocket, the rocket stops immediately.

In addition to what has been said already. "Immediately" means different things in the different frames so you need to specify in which frame it stops "immediately". If the rocket would stop "immediately" in one frame, it would not do that in another frame.

 

[exponent137]

It stops immediately in my frame (of the earthman). I think that this agrees with Ibix answer?

What about the other option that the rocket is stopped immediately in the same some moment according to the rocketer and it goes to the inertial system of the eatrhman? Is something special about the length? I think that the rocket is stretched for the factor $\gamma x' v^2/c^2$, where x'=10 m, thus it is stretched for 30 m instead of 5 m in the above example.

However, it is interesting to me here that the existence of a rigid body here is used, but the answer is correct anyway, according to Ibix.

 

[Ibix]

It stops immediately in my frame (of the earthman). I think that this agrees with Ibix answer?

It does if you program the many engines to fire at the same time as defined by the earthman's frame. Which was what I was assuming, and should have stated explicitly. You could, of course, program the engines to fire simultaneously as defined by any other frame.

What about the other option that the rocket is stopped immediately in the same some moment according to the rocketer and it goes to the inertial system of the eatrhman?

What do you think? The Lorentz transforms will help you here.

 

[exponent137]

What do you think? The Lorentz transforms will help you here.

I answered above, 30 meters if "simultaneous" means for rocketeer.

Your model can be simplified so that we have two small rockets, one at the start of the rocket and one at the end of the rocket. In the same moment, (according to earthman) they stop immediately. The point is that although they are not a rigid or connected body, this is possible. And the distance stays the same as at moving. This is a little easier to visualize.

 

[Dale]

When this point stops immediately, (when the end of the rocket flies close to me) it is immediately transferred 5 m ahead.

This is a self contradiction. If it immediately stops then it cannot move 5 m ahead. You can envision a scenario where it stops immediately, and you can envision a scenario where it continues to travel 5 m, but it is contradictory to say that it does both.

 

[pervect]

A rocket flies past the me with velocity $0,866$ c, therefore $\gamma=2$. Its length in rest is 10 m. When I am parallel with the last part of the rocket, the rocket stops immediately. The last part stays parralel with me, but the beginning of the rocket jumps for factor 2, therefore jumps for 5 meters immediately. This implies that deacceleration should have some limit that it prevent velocity larger or equal than c. But I never heard for such connection between acceleration and speed limit.

One possible objection is that a relativistic rocket does not behave as a rigid body; Lorentz equations do not prevents that all the rockets stops immediately, but except of this $v>c$. Maybe this seems the most logical answer to this paradox.

Where I am wrong, or what a correct answer to this is?

It's not quite clear what properties you are imagining the rocket has. Are you imagining the rocket as rigid? Or are you imagining the rocket as some collection of points, not necessarily rigid, and all the points suddenly stop moving "at the same time"? The two cases are different. To answer the question we need to be able to describe the motion of all points on the rocket. We know that the tail , by the problem statement, suddenly stops, but we need to understand theoretical conditions that the other points on the rocket must satisfy to answer the question of what happens to them, they are not specified by the motion of the single point. You might be trying to leverage off the notion that in Newtonian mechanics that specifying the motion of one point on a rigid 1d body specifies the motion of all the points, but it's not clear if you're imagining the rocket as a rigid body or not.

In the second case (which seems to me to be what you're asking, since you didn't mention rigidity, but I could be wrong), you need (as previously mentioned by others) to define some simultaneity convention, there is no universal notion of "at the same time" in special relativity. You haven't specified one, so if this is what you are asking, my guess would be you aren't familiar with the idea of relativity of simultaneity, or have overlooked the issue.

If you're not familiar at all with the topic, and have somehow never seen it before, I can only suggest you read about it, there is a lot written on the topic, both on and off PF. If you have read the words before, but still "don't get it", I'm not sure what to recommend. If the omission is just a momentary oversight based on old habits, then you are in a bit better position, you can correct the oversight and try to refine your question.

Sorry for the length, and the digressions, but I'm not quite understanding what you're trying to ask, and I hope some discussion of the most likely possibilities of what you might be trying to ask will clarify the question.

 

[exponent137]

Dale, Pervect
Now it is clear for me. I though that length of a moving rocket changes to a length of a rest rocket when rocket stops. But, this is not true in such case, as Ibis explained me.

 

[pervect]

The detailed case of a Born-rigid rocket, with a constant proper length, stopping, is an interesting one. In this case, the whole rocket cannot stop suddenly, but in the appropriate limit, one can make a single point on such a rocket stop suddenly. In this scenario, the single point which stops suddenly has an infinite proper acceleration that lasts for an infinitesimal amount of time, which is consistent with a sudden stop. Other points on this Born-rigid rocket have a finite proper acceleration, which lasts for more than an instant, meaning that they can't be described as stopping "suddenly".

 

[Herman Trivilino]

But, something is even unclear to me:
Let us say that one point flies parralel to the beginning of the rocket. When this point stops immediately, (when the end of the rocket flies close to me) it is immediately transferred 5 m ahead.

If you go back and look at the way you stated your original paradox, you'll see that this is just another version of it. When you observed the back stop, you said the front would move 5 meters. Now you observe the front stop, so you'd say the back moves 5 meters.

If you resolve the first version you immediately resolve the second version.

 

[exponent137]

If you go back and look at the way you stated your original paradox, you'll see that this is just another version of it. When you observed the back stop, you said the front would move 5 meters. Now you observe the front stop, so you'd say the back moves 5 meters.

If you resolve the first version you immediately resolve the second version.

It was a lapse in my way of thinking, but Ibix and others helped to resolve it. I erroneously thought that a rest rocket is always longer than the moving rocket.

But, the second case has a new meaning, because it excludes the paradox with a rigid body.

Regards