Bell's spaceship clarification

[LAHLH]

Hi,

I'm reading the excellent explanation of this paradox at http://www2.corepower.com:8080/~relfaq/spaceship_puzzle.html but have become confused by the wording in this paragraph:

The x'-distance between the curves is actually slightly greater than gamma*k. You can see this by a geometrical construction. Remember that P is point on the left-hand curve, and the t'-axis passes through P and is tangent to that curve. Draw a horizontal line (a line t=constant) through P; let this line cross the right-hand curve at Q. Draw a line parallel to the t'-axis through Q (a line of the form t'=constant). Fact: the x'-axis crosses the two slanted lines, t'=0 and t'=constant, at x'-coordinates 0 and gamma*k, respectively. Since the right-hand curve is tangent to t'=constant at Q, the x'-axis will cross the right-hand curve at x' > gamma*k. [Exercise: rephrase this physically!] [End of bracket convention.]

It could just be me missing something but this paragraph seems contradictory. When he says draw a line parallel to the t'-axis through Q but then in brackets says "a line of the form t'=const", surely these things are opposites? the lines of the form t'=const are perpendicular (in the Lorentzian sense-equal sides of 45 deg null line) to the lines parallel to the t'-axis ? just as in cartesian geometry lines of y=const are perp to the y-axis etc.

So that is my first point of confusion.

Then he talks of the x'-axis crossing the t'=0 line: surely the x'-axis axis is the t'=0 line??

he has totally lost me at this point. The picture I have currently is attached...Just wondering if someone could shed some light on this for me as I was enjoying the explanation up until here.

 

[Bill_K]

LAHLH, You're right, he has several typos, and they are crucial to the discussion.

"Draw a line parallel to the t'-axis through Q (a line of the form t'=constant). Fact: the x'-axis crosses the two slanted lines, t'=0 and t'=constant, at x'-coordinates 0 and gamma*k, respectively."

should be

"Draw a line parallel to the t'-axis through Q (a line of the form x'=constant). Fact: the x'-axis crosses the two slanted lines, x'=0 and x'=constant, at x'-coordinates 0 and gamma*k, respectively."

 

[LAHLH]

Ah, thanks a lot!

 

[LAHLH]

Also the next bit should read

Since the right-hand curve is tangent to x'=constant at Q, the x'-axis will cross the right-hand curve at x' > gamma*k

?

Looking at the Wiki article seems to suggest the distance is = gamma*k, not greater? is this just because the wiki article considers the case after acceleration has ceased (engines turned off)? So effectively the x-axis will cross the curve at x'=gamma*k for this case as the x'=const line remains tangent forever after now the engines are off?

Finally, is this paradox actually settled?

 

[JesseM]

Also the next bit should read ?

Looking at the Wiki article seems to suggest the distance is = gamma*k, not greater? is this just because the wiki article considers the case after acceleration has ceased (engines turned off)? So effectively the x-axis will cross the curve at x'=gamma*k for this case as the x'=const line remains tangent forever after now the engines are off?

Yes, I think they're assuming we consider the distance in both frames after the ships stop accelerating.

Finally, is this paradox actually settled?

Yes, it's understood the rope would eventually break.

 

[Meir Achuz]

You may want to look at <http://arxiv.org/abs/0906.1919>.