- #36

yuiop

- 3,962

- 20

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I saw that you talked about two people jumping from different floors of the same building earlier in this thread. Your argument seems to be that since the distance between those guys (as measured by one of them) won't increase, even if we pretend that the pull of gravity doesn't vary with altitude, the distance between the rockets (as measured by one of the rockets) won't either. I would reverse that argument and say that since my posts above prove in a very simple way that the string must break, we are forced to conclude that the distance between the two guys in free-fall must also increase.

The distance between the two guys in free-fall in a uniform gravitational field will

**not**increase.

The reason is this:

2 observers (A and B) that are in free fall in a uniform free fall (after jumping off a tall building is equivalent to 2 inertial observers (C and D) far out in space far away from any large gravitational body that are at rest with respect to each other and watching a tall but massless building being accelerated past them by a rocket.

Observers (C and D) do

**not**see the distance between them as changing with time and any string joining them will

**not**be stretched and the same is true for observers A and B. It is is also true that observers A and B and observers C and D do

**not**feel any acceleration. They only see it when they look at the building.

2 observers (E and F) that are accelerating with constant equal acceleration according to an inertial observer are not equivalent to the situation of observers A and B and nor are they equivalent to the situation of observers C and D and this is made clear when it is noted that observers E and F feel acceleration and would know the difference even with their eyes closed.

2 observers (G and H) that are on separate floors of a tall building located in a gravitational field (and at rest with their respective floors) are in an equivalent situation to observers (J and K) that are in a massless building far away from any significant massive body that is being accelerated artificially. An inertial observer watching the building being accelerated in space sees the building as length contracting as it accelerates. Observers G and H in the gravitational field obviously do not see the distance between floors of their building as increasing over time. Because the situations of observers G and H and J and K are equivalent it is also obvious that observers J and K do not not see their separation as increasing over time. Neither the situation of observers G and H and J and K are are equivalent to that of observers E and F even though all thee pairs of observers actually feel acceleration.

The nearest equivalent to the situation of observers E and F is that of two observers in a building being artificially accelerated and being artificially expanded at the same time, or two observers on different floors of a tall building in a gravitational field where the building is getting rapidly taller over time. Observers E and F are equivalent to the classic situation described in Bell's paradox and Bell's paradox can not be compared to the situation of observers (A and B) or (C and D) or (G and H) or (J and K) as described above because none of those situations are equivalent.

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I solved the original problem completely in my first two posts in this thread, so I encourage you to take a look at them again and try to find something wrong with my argument. l also recommend that you draw a space-time diagram.

You could always refer to the diagram I posted in post#1 of this thread which has the paths and points in spacetime accurately drawn using geometrical software with coordinates tranformed using the the Lorentz transformations.

What if the Earth accelerated right along with them as soon as they launched? Would the string still break?

The original Bell's paradox does not include the Earth as a gravitational body but just as a point of reference. As in the twins paradox the Earth is not meant to represent a source of acceleration and is loosely used as inertial reference frame even though it is not in reality. In the though classic thought experiments, the Earth is imagined to be an ideal massless point of reference with no significant gravitational field. As such it would make no difference if the Earth accelerated right along with accelerating rockets. The rockets are only required to maintain constant proper acceleration which they can measure without even looking out of a window by using onboard accelerometers. If the Earth is replaced by a small spacestation it should be clear that the spacestation accelerating after the rockets have accelerated would make little difference to the proper acceleration measured by the onboard rocket accelerometers.