1. Jan 8, 2012

### riverway

Hi,

I am trying to understand this Bell's Spaceship Paradox , which is basically asking when there is a thread connected between two spaceship (A,B) and an observer C observes a "constant" acceleration, then whether this thread will break or not by Lorentz Contraction.

I think the problem is a little bit ill-defined in Crowell's book. But what if we refine the question as like this:

1) At t=0, the two spaceship (A, B) and the observer C was at rest.
2) A & B started to accelerate at t=0 with "constant" acceleration in the sense that they both engaged the same engine power (as defined in D'Inverono's "Introducing Einstein's Relativity" p.37).

Then, as in D'inverono's deduction, the location of two spaceships after t can be written as :
(x-x0) = (c/a) sqrt(c^2 + a^2(t-t0)^2)-c^2/a

From this, I think the separation between A & B will remain constant all the timeand there is no reason that the thread connecting A & B should break. But this (my) conclusion is the opposite with Bell's and Crowell's argument.

Crowell says "By the equivalence principle, any experiments done by A and B give the same results as if they were immersed in a gravitational field. The leading ship B sees A as experiencing a gravitational time dilation. According to B, the slowpoke A isn’t accelerating as rapidly as it should, causing the string to break." It sounds persuading too.

Could any one help me to understand this discrepancy?

Last edited: Jan 8, 2012
2. Jan 9, 2012

### ghwellsjr

You are correct, the separation between A & B will remain constant all the time in the frame in which C is at rest, but the spaceships and the thread will contract along the direction of motion and so the thread will break.

3. Jan 9, 2012

### Passionflower

Correct, end of story!

(but I suspect this to become another 200 page thread)

4. Jan 9, 2012

### riverway

I am a newbie in Physics Forum here and I searched threads before posting. I used keywords of "Bell Paradox" and the result seemed like thousands of random posts! As soon as I posted my question, a list of very long replied previous posts related Bell paradox were shown right away. I didn't know this topic was so thorughly discussed in this forum. Forgive me but I like to share my emabarassment with the poor search logic of this forum ;-)

So, I quckly went through those posts and got some resolution now. The comoving frame of A or B (or thread) is not an inertial frame any more as soon as it starts. I mistakenly thought that as far as the distance between A & B is constant in C frame, the Lorentz Contraction will happen uniformly thoughout [A - Thread - B] system and thread shouldn't feel any tension. That would be true if [A-Thread-B] is moving in constant velocity but it's not when accelerated.

Thank you guys, anyway.

5. Jan 9, 2012

### Fredrik

Staff Emeritus
vBulletin's search feature is really bad. I suspect that it didn't even include "Bell" in the search because it's too common or too short or something.

6. Jan 9, 2012

### ghwellsjr

If you go to the advanced search and select Search Titles Only and search just in this forum on "bell", you get almost the same threads that appear at the bottom of this page.

7. Jan 9, 2012

### harrylin

As gravitational effects were inferred from SR based on the equivalence principle, reversing that isn't needed and even counterproductive, IMHO...

Did you try the Wikipedia article? It's also not very good but at least it directly refers to the original articles, and the intro is rather straightforward I think. I think that the argument of Dewan and Beran directly kills your argument.

8. Jan 9, 2012

### Staff: Mentor

Another useful technique is to use the "Search" link at the top of the page, and choose the "Search PF via Google" option. This allows you to search for an exact phrase by using quotes, e.g. "Bell spaceship paradox":

The drawback is that you can't restrict the search to a specific forum as you can with the forum's built-in search engine.

9. Jan 12, 2012

### riverway

It took a few nights to read (or just some quick glance) all those postings in this forum. And yes, I read Wiki's article about "Bell Spaceship Paradox". Its argument and drawing was quite clear to understand why the tread will break.

Among all of them, Fredrik's argument was approaching most succintly, at least, to me. The fact that the observer C (stationary) sees the same distance between two ships all the time in spite of the Lorentz Contraction means that the distance btw ships in ship's comoving frame must have increased. The observer C will see two shortened ships and a constant distance btw those ships.

At rest, in everyone's frame including C:

>--------> . . . . . . >-------->

After acceleration, in C's frame:

>--> . . . . . . >-->

Another fresh air came from "Proper Acceleration" and "Rigid Motion" in Rindler's book, "Relativity". To acheive a rigid acceleration, the different part of a thread (or rod, spaceship whatever) must accelerate in uniformly different rate in its length, in the way that rear-end must accelerate faster than front-end. This is obvious when you see the world line diagram with hyperbolas (which means constant proper acceleration motions). This indicates that the same acceleration of two ships cannot maintain the distance between them in one of their frames. It will increase.

harrylin:
In defence of Crowell, the cited part in my first posting came from his book "General Relativity". So the paradox was explained in GR context. And besides, that specific argument was the third one after two SR arguments in his text.

One said "Right is right in only one way but wrong is wrong in million different ways." Now, I cleared one of my confusions and million minus one to go.

Thank you guys. You guys were terrific.

Last edited: Jan 12, 2012
10. Jan 12, 2012

### ghwellsjr

Fredrik's argument must be so succinct that I can't even find it. Was it in one of the links or searched threads?

11. Jan 12, 2012

### riverway

If you look at the bottom of this page, there is a post "Why is the Wikipedia article about Bell's spaceship "paradox" disputed at all?". Fredrik's argument is the first posting of it.