Sorry, I always INTEND to specify which frame I'm talking about. I suspect that usually I meant that the frame is that of the people on the rockets.
Yet again, the rockets are not inertial, so a reference to them does not identify a frame.
You seem to have indirectly conceded that your 'proof' has errors in it, and its conclusion is faulty, if nothing else by the fact that you incorrectly say that Bell's scenario is irrelevant, and yet it drags in Bell's scenario. You're contradicting yourself.
So I will attempt to fix some of the errors (in red, the way you like it. The first one references only frame S in which Alice is always stationary,
First of all, we're going to reference the case where he's already out there, since it just isn't Bell's unless they're relatively at rest at first, and Alice's rocket undergoes the same acceleration as does Bob's. This isn't true at all if there's an abrupt turnaround without a pause, until he's 30 in this case.
A Constant Separation of the Rockets in S does not Contradict anything
During the traveler's (his) instantaneous acceleration homeward when he turns 30, his home twin's (Alice) age of 50 in S is unchanged. But IF it's true that the two separated rockets in the Bell's Paradox (whose accelerometers show equal constant readings) DON'T maintain a constant separation in S during Bob's acceleration, that CONTRADICTS Alice's empirical observations.
Here's how to see that contradiction:
Bob's acceleration duration is zero, so in any inertial frame, including that of S, the duration of Alice's worldline is also zero. Therefore she should experience no change in location of the ship in her vicinity during that period of zero time. THAT would result in HER seeing the leading rocket INSTANTANEOUSLY move a finite distance away from her, WHICH IS ABSURD! So the ASSUMPTION that the separation of the rockets in S isn't constant during that one moment in time CAN’T be correct.
I will do it again, this time using the S" frame in which the rockets are stationary after their respective acceleration events:
A Non-Constant Separation of the Rockets in S" does not Contradict anything
During the traveler's (his) instantaneous coming to a halt in S" when he turns 30, his home twin's (Alice) age of 80 in S is unchanged, so similar to above, Alice notices no teleporting of anything at any time. The two separated rockets maintaining a constant separation in S" would contradict relativity of simultaneity.
Here's how to see that contradiction:
In frame S, the two rocket accelerations are simultaneous. Thus in frame S" (or any other frame in which the x component of pre-acceleration velocity is nonzero), those two events cannot be simultaneous. If they're not simultaneous in S", then one acceleration occurs before (in S") the other, and that must change the separation distance in S.
In this particular instance, Alice's ship stops (in S") first, and Bob's ship stops (in S") 60 years (in S") later.
My apologies, but most of the text got removed since it was all either wrong or irrelevant. There's just no inertial frame (or any other kind of frame for that matter) where the worldline of any object (person, rocket, clock, whatever) is discontinuous. You agree with this because you drew the same picture as us, at least for this abrupt acceleration case. Your conclusion seems to rely on such a discontinuity, which has just not been demonstrated, mostly due to you fooling yourself by your own lack of frame references.
Either wording shows that a string stationary in S connecting the rockets while the rockets are stationionary in S would be of length 34.6. It also shows that after the acceleration events both take place, a string stationary in S" connecting the rockets after the rockets both become stationary in S", would be longer.
Mike_Fontenot said:
I had said:
"the widely-held (but incorrect, according to me) view that two separated rockets with equal accelerometer readings will get farther apart as the acceleration proceeds"
And Halc responds:
Nobody who knows relativity would say that, so it is a stretch to call it a 'widely held view'.
Mike responds:
I thought that IS what you believe.
It, lacking a frame reference, is meaningless, and nobody who knows their relativity would make such a frame-dependent statement sans frame reference. They maintain constant separation in S, so your comment as worded is incorrect, and only utter novices would say such a thing. There are a lot of utter novices, so I didn't deny the truth of the comment, I just said that those who know their relativity wouldn't have said that.
The string won't break if the separation of the rockets (according to the people on the rockets) is constant.
For that to happen, the front rocket would have to have less acceleration in order to not be moving faster than the rear rocket in the inertial frame of either rocket. A denial of that is a denial of relativity of simultaneity. If they're constantly accelerating and moving at the same velocity in S, they can't be moving at identical velocity in another frame. That would imply that the velocity is the same at two different times, contradicting the fact that each is supposedly accelerating.
The people on the rockets certainly have their own frame.
Since all people and other objects are in every frame, any particular frame is not necessarily 'theirs'. Almost nobody uses such a convention. When I get in my car, I consider myself to be going fast, not the road. That's me choosing to use a frame other than the one in which I am stationary.
Specify the frame, especially since the people are not stationary, so saying 'Bob's frame' does not identify an particular frame at all. It's easier and far clearer to say 'in S' than it is to say 'according to accelerating Bob'.
When they say "the separation between our two rockets is constant", they are NOT talking about ANY inertial frame.
Obviously not, since no frame is mentioned. So be better than that and mention the frame.
There are many forms of accelerating coordinate systems. Saying 'Bob's accelerating frame' does not identify which of those systems you're using, or which Bob is using. For instance, you have zero clue which coordinate system Einstein was using when he wrote about that equation that you called the GTD. He was using Lass coordinates, which makes sense since it most closely resembles the way simultaneity conventions are used in his earlier works.
Yes, if Bob switches abruptly from using S to using S" (something he's free to do anytime, not requiring acceleration at all), then he changes his LoS to a different event on the lead rocket's worldline. It's a change in abstraction only, not a physical change to the worldline of the rocket. But your 'proof' (omitting all frame references) totally misses that.