B Bell's Spaceship Paradox & Length Contraction

[liuxinhua]

Each spaceship has its own original time τ.
For spaceship A at its every original time, such at time τi, there exists an inertial reference frame Ki spaceship A and spaceship B is rest in Ki. When the original time of spaceship A is τi , corresponding time in Ki is ti.
For spaceship A at time τj, there exists an inertial reference frame Kj spaceship A and spaceship B is rest in K. When the original time of spaceship A isτj , corresponding time in Kj is tj.
Measured in Ki at time ti and measured in Kj at time tj, the acceleration of spaceship A is a constant.
The distance of spaceship A and spaceship B is l₀ measured in Kj at time tj.

Thank for stevendaryl.

 

[Mentz114]

You and two identical spaceships are all at rest with respect to each other. You note that the two engines start up at the same time, and the thrust curve and acceleration profile of both spaceships are identical. As the ships pick up speed, would you measure the ships to be shorter than their rest length?

These diagrams show that from any instantaneously comoving frame (ICF) on any ship the trailing and leading ships are receeding and the strings must break .
The first diagram is at time zero with all clocks synchronised at $\tau=0$.
The next diagram shows the frame that is comoving with the leftmost clock at its time ≈ 8. This is the top white line. This frame is also comoving with each ship at $\tau$≈8. For each white line the ships to the left of the blue square have negative velocity and those to the right have positive velocity

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https://www.physicsforums.com/attachments/229366

 

[David Lewis]

…shouldn't we consider an engine that applies the thrust uniformly across the entire ship??

I don't think so in this case. The engine (as I've drawn it) applies force just to the rear of the ship. The ship will compress slightly due to classical mechanical deformation but I am ignoring this and for simplicity just focusing on relativistic effects.

The only way for the string not to break is if the car/rocket in the rear is accelerating a tiny bit more than the car/rocket in the front.

You're probably right. Consider, however, that the acceleration of both ships is the same. For the observer (who was initially at rest with respect to the ships) both ships started moving at the same time. For the men in the ship, the front ship got a head start.

 

[sweet springs]

You're probably right. Consider, however, that the acceleration of both ships is the same. For the observer (who was initially at rest with respect to the ships) both ships started moving at the same time. For the men in the ship, the front ship got a head start.

Same, you say, should be investigated more in detail.
The two engines are the same product and in the same condition. The two pilots are well trained to keep the same starting manual.
The two pilots and a commander on the Earth share the same IFR when the pilots fire engines of the rockets on the Earth or staying still in space with the Earth. The rockets start at the same time for all the three. No head start.

For the commander all the things of the rockets keep same during the flight, constant distance, the same speeds and the same rocket lengths at his Earth time, etc. However, though the same start,
In the front rocket FR, the rear rocket's engine is in sooner phase in accerelatin flight manual than his own. "The rear pilot is reading chapter I of manual though I am reading chapter II"
In the rear rocket FR, the front rocket's engine is in later phase in acceleration flight manual than his own. "The front pilot is reading chapter II of manual though I am reading chapter I"
The two pilots share the same judgement in synchronisity, "the more front, the more future". They observe that the distance between the two strats increasing. So the thead is torn. The distance would be shortened in latter phase for reducing power for inertial flight in space. After all the starting procedures are completed by the with engines cut, the distance of rockets is the same initial value for the three.

In order that the thread is not torn apart the front pilot should reduce power, the rear pilot should increase power or the both is required. The different manual should be given to the pilots for the mission of "no cut of thread during the flight".

 

[stevendaryl]

You're probably right. Consider, however, that the acceleration of both ships is the same. For the observer (who was initially at rest with respect to the ships) both ships started moving at the same time. For the men in the ship, the front ship got a head start.

Not just a head start. From the point of view of the passengers on the ships, the distance between the ships keeps growing. So unless the string can stretch infinitely far, it will break.

Let $L$ be the distance between the two ships as measured in the "launch" frame (where they are initially at rest). Now, wait a while until the ships are traveling at speed $v$. Now, transform to a frame in which the rear ship is momentarily at rest. In this frame:

• The rear ship is (momentarily) at rest.
• The distance between the ships is greater than $\gamma L$
• The front ship is getting farther away from the rear ship.

So whatever is the maximum the string can stretch, eventually $\gamma L$ will get bigger than that, and the string will break.

 

[stevendaryl]

Another way to look at the string breaking is in terms of the Rindler horizon. Consider a rocket ship that is traveling according to the trajectory:

$x_{rocket} = (x_0 - \frac{c^2}{g}) + \sqrt{\frac{c^4}{g^2} + c^2 t^2}$

(that's constant proper acceleration $g$ starting at rest at $x=x_0$ at time $t=0$)

Then there is a second trajectory behind the first:

$x_{horizon} = x_0 + ct$

For all $t$, $x_{horizon}(t) < x_{rocket}(t)$.

If you have an observer whose trajectory is such that $x_{observer}(t) < x_{horizon}(t)$, then there is no way for that observer to send a signal to the rocket. That's sort of obvious, because $x_{horizon}$ moves at speed $c$, so something that gets behind the horizon will never be able to catch up with it again.

In the two rocket case, the rear rocket will fall below the Rindler horizon of the front rocket.

 

[David Lewis]

In the initial launch frame, would it be accurate to say the tension in the thread increases as the ships pick up speed because the molecules of which the thread is composed flatten in the direction of motion?

 

[PeterDonis]

In the initial launch frame, would it be accurate to say the tension in the thread increases as the ships pick up speed because the molecules of which the thread is composed flatten in the direction of motion?

No. The shape of the molecules plays no role, since, as pointed out in post #55, we are talking about SR, not quantum mechanics. In SR, the thread is assumed to be a continuous line extending along the direction of motion--a real thread has a thickness, but the thread's extent in the directions perpendicular to the direction of motion is irrelevant to the SR analysis.

 

[A.T.]

In the initial launch frame, would it be accurate to say the tension in the thread increases as the ships pick up speed because the molecules of which the thread is composed flatten in the direction of motion?

The fields that hold the string atoms together are contracted in the initial launch frame, to they produce more attractive force, than in an identical string at rest in the that frame, despite the fact that both strings remain at equal length in the frame.

 

[vanhees71]

You can find some calculations concerning the space-ship paradox in my SRT article here (p. 33):

https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[David Lewis]

No. The shape of the molecules plays no role, since, as pointed out in post #55, we are talking about SR, not quantum mechanics.

Then I don't understand how an object can get shorter.

 

[pervect]

Then I don't understand how an object can get shorter. View attachment 243696

If molecules had a shape, they'd get shorter. In the world of quantum mechanics, though, it's not clear if molecules actually have shapes or not, at least not to me. I would say that molecules have wavefunctions which occupy a non-physical "configuration space", with 3 dimensions for every particle in the molecule (presumably these particles are atoms, but you could break the atoms down into more particles). That's not really a "shape" as far as I am concerned.

Most treatments of introductory QM treat single particle systems, where the wavfunction does occupy normal space. It's when one considers multi-particle systems that one gets into the issue of the wavefunctions not occupying physical space.

But it's much simpler to keep the arcane aspects of QM out of the discussion, which is what the original point was.

 

[PeterDonis]

Then I don't understand how an object can get shorter.

Because that's how the geometry of spacetime works. The SR treatment of the Bell spaceship paradox does not make any hypothesis about the internal structure of the object. It just explores the consequences of the stated conditions, given the geometry of Minkowski spacetime. This geometry puts constraints on any model of an object's internal structure; but it doesn't tell you anything specific about that internal structure.

 

[PeterDonis]

I don't understand how an object can get shorter.

As far as an "object" describable by classical physics is concerned, Lorentz showed in the 1890s (IIRC) that any object made of electric charges bound together by electromagnetic fields would exhibit length contraction in a frame in which it was moving.

 

[A.T.]

Then I don't understand how an object can get shorter.

The string in Bell's scenario doesn't get shorter, so the contracted binding EM fields have to span the same distances. Hence the tension. To avoid the complications of QM don't go down to the atomic level, but instead consider the contracting links of a chain that is forced to keep a constant length.

 

[pervect]

The string in Bell's scenario doesn't get shorter, so the contracted binding EM fields have to span the same distances. Hence the tension. To avoid the complications of QM don't go down to the atomic level, but instead consider the contracting links of a chain that is forced to keep a constant length.

I agree. If one has a rod or string which one divide into classical pieces of matter, then, when the rod undergoes length contraction, so does each of the pieces of the rod.

It doesn't matter to the argument how small each of the pieces is.

It does matter to the argument that we consider the pieces to behave classically. It's unclear to me how one rigorously deals with the quantum aspects, but this argument can go in another forum such as the quantum forum. I would guess that there is some sort of classical limit one can take, but I've never seen a serious formal discussion of the issue. This doesn't mean that one may not exist, as I'm not too familiar with the appropriate literature, unfortunately.

 

[PeterDonis]

It's unclear to me how one rigorously deals with the quantum aspects, but this argument can go in another forum such as the quantum forum.

Exactly.