Twin paradox for (accelerated) dummies?

[Dale]

The metric in this case doesn't change between the two trajectories and it's the same in both, because when g is towards the turning point, the x'-coordinate that Bob considers is positive, but when the spaceship inverts acceleration (let's say "-g"), then the x'-coordinate is negative, because it's then referenced to a turning point in the future, so it's at the negative x'-side of the new trajectory.

This doesn't work. The Rindler frame is for a constant acceleration, it is not valid for a varying acceleration. Flipping the coordinates is a clever idea, but then it is not a single chart but multiple charts with a coordinate transform (e.g. $x=-x'$) between the two charts.

Either approach is viable, but as written you are incomplete either way. Either you need to revise the metric for a single chart or you need to identify the transform between the multiple charts you are using.

Note, this is not a negative reflection on your work here. For new posters this is far better than anyone else I have seen in a very long time. It is just letting you know some of the difficulties that make doing this sort of thing rigorously more difficult than many people appreciate.

 

[member 728827]

In other words, she is in free fall the whole time. That's what I was asking.

Ok, so for Bob perspective Alice is in free falling the whole time. Then at the start position in Earth at T=t'=0, when Bob jumps into the spaceship, in Bob's non-inertial reference Alice is in free falling and is moving AWAY from Bob, but after some time after the acceleration switch event, Alice begins "free falling" TOWARDS Bob. Seems a Sesame street clip.

So is always "free falling", but changed from "going away" to "coming towards", what is not something you can explain changing the sign of the x'-coordinate.

If you mean that "free falling" means that the only force that acts over Alice is "gravity", or using GR equivalence principle an acceleration, then neither is true, because the "net" acceleration changed its direction, and during that transient phase, even when Bob really feels the effect of the change, will describe what is happening to Alice as a new force acting on her, that was not there before. You can think that the original "cause" of the "+g" acceleration is still there, but now there's a new "cause" that produces "-2g".

And is not free falling by any definition you try to apply. It's quasi-quasi-free falling, but not quite, that was what I answered.

 

[Ibix]

And is not free falling by any definition you try to apply. It's quasi-quasi-free falling, but not quite, that was what I answered.

No, it's always free falling by the only measure that actually matters: what do accelerometers attached to Alice show? They always show zero. Thus she is in free fall - that is the definition. Bob trying to interpret her motion in terms of time varying gravitational fields does not change that her weighing scale always reads zero.

 

[member 728827]

This doesn't work. The Rindler frame is for a constant acceleration, it is not valid for a varying acceleration.

The Rindler frame is derived for the infinite series of attached CMRF that at every instant has the same velocity as the non-inertial frame, and the "clock-hypothesis". If that hypothesis is true, the patch applied to transit from one worldline with +g, to the other with -g if there's continuity in the speed of the MCRF is correct.

 

[Ibix]

The Rindler frame is derived for the infinite series of attached CMRF that at every instant has the same velocity as the non-inertial frame, and the "clock-hypothesis". If that hypothesis is true, the patch applied to transit from one worldline with +g, to the other with -g if there's continuity in the speed of the MCRF is correct.

Yes, what you've done works, I think (not sure off the top of my head, but you might need to worry about discontinuity in your X coordinates - however that's an annoyance rather than a problem). I think @Dale's point is that it's not one Rindler coordinate system with the sign of $g$ flipping, it's two coordinate systems with opposite $g$s stitched together.

 

[PeterDonis]

for Bob perspective Alice is in free falling the whole time

The fact that Alice is free-falling the whole time is an invariant; it's true regardless of anyone's "perspective".

at the start position in Earth at T=t'=0, when Bob jumps into the spaceship, in Bob's non-inertial reference Alice is in free falling and is moving AWAY from Bob, but after some time after the acceleration switch event, Alice begins "free falling" TOWARDS Bob.

Yes, because you've changed the frame you're using. Nothing changed about Alice.

changed from "going away" to "coming towards", what is not something you can explain changing the sign of the x'-coordinate.

Yes, you can; in fact that's exactly the "explanation" for the "change"--because, as above, nothing about Alice changed. You can't magically change what Alice is doing by changing the way Bob is describing things. All you can do is change the way Bob is describing things, and that's what the sign change in the x coordinate does. It's purely a change in description.

The only actual physical change is that Bob changes the direction of his proper acceleration, from "away from Alice" to "towards Alice". And he chooses to change his description (what frame he is using) at the same time that he makes this physical change. But you still need to be careful to keep distinct the physical change in the direction of Bob's proper acceleration, which is an invariant, from the change in Bob's description, which is not--there is no corresponding change in Alice's description using the inertial frame in which she is at rest the whole time.

 

[Dale]

After some mentor deliberation, this thread will remain closed