Is the Bell's Spaceship Paradox a Result of Relativity of Simultaneity?

[Trojan666ru]

In relativity of simultaneity, it is said like simultaneous
events for one observer is not simultaneous for another.
Does that mean events been seen from one observers
point of view may not be true?
For example consider the bell’s spaceship paradox
From my point of view the "silk thread" which was tied
between the two spaceships were broken because of length
contraction. But from the travelers point of view they
never feel any length contraction or any weird effects, they are the same as if they were in rest so the silk thread won't break in their view. Later they'll return to Earth and we notice the silk thread remain unbroken. So point of view was true? Travelers or observers?

 

[dvf]

But from the travelers point of view they
never feel any length contraction or any weird effects, they are the same as if they were in rest so the silk thread won't break in their view. Later they'll return to Earth and we notice the silk thread remain unbroken. So point of view was true? Travelers or observers?

From the travelers point of view they don't accelerate the same way. The one in front moves away from the one behind. The thread breaks.

 

[yuiop]

If the thread breaks according to any observer, then it breaks according to all observers.

If the rear rocket accelerates faster than the leading rocket in such a way that the distance between the rockets remains constant according to the observers on the rockets, then the thread does not break according to any observer. In this situation, the gap between the two rockets gets progressively shorter according to an inertial observer in accordance with length contraction and so this observer does not expect there to be any stress on the thread. If the rockets accelerate in any other way, for example keeping constant distance between them as measured by an inertial observer, or maintaining equal constant proper acceleration or maintaining equal instantaneous velocity relative to a given inertial reference frame, then the thread will break and all observers will agree on this.

In relativity, different observers may disagree on the temporal order of spatially separated events, but never disagree on single events that occur in one location, such as a thread breaking or a bomb exploding, etc.

 

[Trojan666ru]

Lets imagine from the spaceships point if view. What i really don't understand is how do the thread break if both the spaceships are accelerating at same speed? accelerating At same same speed means they are keeping their distance between them the same, or not? if not how?

 

[dauto]

They don't accelerate at same speed. They accelerate at same acceleration - duh, but one starts accelerating before the other and will move faster than the other breaking the thread.

 

[Nugatory]

Lets imagine from the spaceships point if view. What i really don't understand is how do the thread break if both the spaceships are accelerating at same speed? accelerating At same same speed means they are keeping their distance between them the same, or not? if not how?

When you say they are accelerating "at the same speed", you're saying that they are both changing their position and speed at the same time... If one of them turns up the power of his rocket motor the other will do the exact same thing at the same time to keep them moving in lockstep... and because of the relativity of simultaneity, any time you see the words "at the same time", you need to be very careful.

Consider the situation when both ships are moving at a speed $v$ relative to the ground-based observer at the same time. At that moment in the ground observer's frame, the rocket engines of both are firing, increasing the speed of both ships relative to the ground from $v$ to $v+\Delta{v}$. But these two speed increases are not simultaneous in either spaceship frame because of relativity of simultaneity - one ship changes its speed relative to the ground at a different time than the other and therefore the ships move apart in that frame.

This problem is much easier to analyze if you try drawing a spacetime diagram showing the paths of the two ships through spacetime. There are some good ones in this forum, in past threads on this paradox.

 

[A.T.]

Lets imagine from the spaceships point if view. What i really don't understand is how do the thread break if both the spaceships are accelerating at same speed? accelerating At same same speed means they are keeping their distance between them the same, or not? if not how?

They keep their distance in the initial inertial rest frame. They increase their distance in their accelerating rest frames.

 

[Ibix]

In any frame where their separation is constant, there will be length contraction which causes the string to break because it is no longer long enough to stretch between the ships.

In any frame where the separation is not constant, it is increasing. Therefore the string breaks.

Hint: idealisations (like "a massless strong string") that are useful in many situations don't really work here. The masslessness isn't important, but an infinitely strong string has an infinite speed of sound - in violation of the precepts of relativity. You really just have to do the maths...

 

[Dale]

Lets imagine from the spaceships point if view. What i really don't understand is how do the thread break if both the spaceships are accelerating at same speed? accelerating At same same speed means they are keeping their distance between them the same, or not? if not how?

Yes, there is a reference frame where the distance between remains the same at all times.

However, even with a constant distance there is a non-zero expansion tensor. That means that there is a physical strain in all reference frames. It is this strain, due to the expansion tensor, which causes the thread to break in all frames.

 

[harrylin]

[..]
For example consider the bell’s spaceship paradox
From my point of view the "silk thread" which was tied
between the two spaceships were broken because of length
contraction. But from the travelers point of view they
never feel any length contraction or any weird effects, [...]

Hi welcome to Physics Forums!

There is an rather popular approach which is often labeled "accelerating frame", and which consists of continuously making infinitesimal Lorentz transformations to the next instantaneous inertial co-moving frame. That produces a - rather artificial - "accelerating point of view" based on the physics of an infinite number of inertial frames.
But contrary to what you think, the use of such an accelerating point of view for describing physical phenomena produces all kinds of weird effects in conflict with the laws of physics which were defined wrt inertial frames. For starters, it appears as if at take off, with a small amount of rocket fuel, you jerk the whole universe into an acceleration that started back in time for halve the universe (the light of stars that are light years away instantly Doppler shifts). The appearance that the rockets are slowly drifting apart is much less mind boggling IMHO.

The simple solution to getting rid of such paradoxes is straightforward: stick to inertial frames for the physical interpretation. The description from any inertial frame will predict that it appears for the traveler as if the spaceships are drifting apart. Only the exact reason why is disagreed between inertial frames, because of difference in clock synchronizations (relativity of simultaneity).

 

[harrylin]

[..] In any frame where the separation is not constant, it is increasing. Therefore the string breaks. [..]

Sorry but that can't be right: according to reference systems that are moving in the opposite direction, the rocket that is behind will take off first. That implies that the separation is decreasing in such frames. Thus it's a bit more complex than you put it; while in one frame the separation is constant (the easiest frame for calculation), in all inertial frames it's the total effect of distance change and length contraction that matters.