- #31
- 10,465
- 1,627
I've been finding this paper a bit hard to follow, and been busy to boot, so I've been keeping out of it. But I think I may have some comments. And a couple of questions.
I'll throw some things out, I'd like to see if everyone (esp. one of the original authors, who apparently is posting here) agrees with them.
1) There seem to be about five notions of distance that are mentioned in the paper or in this thread.
1a)There are two inertial frames corresponding to the front of the rocket and the back. These have different notions of simultaneity and hence define different notions of distance. These are mentioned near the start of the paper.
We can conclude that in either of these frames, the string breaks when the front and rear rocketships have the same proper acceleration.
1b) There are 2 non-inertial frames which I didn't see mentioned in the paper (I could have missed them), but think they are important enough to deserve special mention and have been mentioned in this thread. These would be the Rindler frames associated with the front of the rocket, and with the rear. By "Rindler frame", I mean the frame associated with a metric of the form
ds^2 = -(1+gt)^2 + dx^2 + dy^2 + dz^2
In the case of equal proper accelerations of the front and rear rocket, these are different frames, and in both of them the string breaks.
1c) The fifth case is the case analyzed in the paper, where the notion of distance is clearly stated to be measured by the hypersurface orthogonal to the congruence of worldlines of the string. I'm not sure if it was explicitly stated in the paper, but I _assume_ that all particles in the string have a constant proper acceleration (?).
This has a different notion of simultaneity than 1a) or 1b) and hence defines a different notion of distance.
One thing I'd like to make sure everyone agrees with: the string always breaks, there is nothing in the analysis to indicate that it does not break.
Another comment. I believe that the notion of distance in this "medium" frame 1c) is inherently incompatible with the notion of Born rigidity - basically, because the "medium" is stretching, objects at rest with respect to the medium are not maintaining a constant distance from each other as is required by the Born notion of rigidity. The other notions of distance 1a) and 1b) have the property that objects at rest with respect to the frames are in Born-rigid motion.
One last point puzzles me. The authors mention that the hypersurface is spatially curved, and that this is important to measuring the distance. This would seem to me to only be important in the case where the underlying geometry is not one dimensional. I.e. if we consider two rockets moving in some Minkowskian frame where y=z=0 for both rockets, we can describe the rockets by their x coordinates. There will be no spatial curvature in the x frame, because you can't have curvature in one spatial dimension, and therefore we won't need to consider the spatial curvature. Only if we had y or z nonzero and different for the front and rear rockets would we have to worry about spatial curvature.
I'll throw some things out, I'd like to see if everyone (esp. one of the original authors, who apparently is posting here) agrees with them.
1) There seem to be about five notions of distance that are mentioned in the paper or in this thread.
1a)There are two inertial frames corresponding to the front of the rocket and the back. These have different notions of simultaneity and hence define different notions of distance. These are mentioned near the start of the paper.
We can conclude that in either of these frames, the string breaks when the front and rear rocketships have the same proper acceleration.
1b) There are 2 non-inertial frames which I didn't see mentioned in the paper (I could have missed them), but think they are important enough to deserve special mention and have been mentioned in this thread. These would be the Rindler frames associated with the front of the rocket, and with the rear. By "Rindler frame", I mean the frame associated with a metric of the form
ds^2 = -(1+gt)^2 + dx^2 + dy^2 + dz^2
In the case of equal proper accelerations of the front and rear rocket, these are different frames, and in both of them the string breaks.
1c) The fifth case is the case analyzed in the paper, where the notion of distance is clearly stated to be measured by the hypersurface orthogonal to the congruence of worldlines of the string. I'm not sure if it was explicitly stated in the paper, but I _assume_ that all particles in the string have a constant proper acceleration (?).
This has a different notion of simultaneity than 1a) or 1b) and hence defines a different notion of distance.
One thing I'd like to make sure everyone agrees with: the string always breaks, there is nothing in the analysis to indicate that it does not break.
Another comment. I believe that the notion of distance in this "medium" frame 1c) is inherently incompatible with the notion of Born rigidity - basically, because the "medium" is stretching, objects at rest with respect to the medium are not maintaining a constant distance from each other as is required by the Born notion of rigidity. The other notions of distance 1a) and 1b) have the property that objects at rest with respect to the frames are in Born-rigid motion.
One last point puzzles me. The authors mention that the hypersurface is spatially curved, and that this is important to measuring the distance. This would seem to me to only be important in the case where the underlying geometry is not one dimensional. I.e. if we consider two rockets moving in some Minkowskian frame where y=z=0 for both rockets, we can describe the rockets by their x coordinates. There will be no spatial curvature in the x frame, because you can't have curvature in one spatial dimension, and therefore we won't need to consider the spatial curvature. Only if we had y or z nonzero and different for the front and rear rockets would we have to worry about spatial curvature.