"NosyNed" writes:
I've always taken it that GR would be used to show this "jump". Is that correct?
Yes, pretty much. There is a way you can make Special Relativity treat it, but in reality it is a GR question.
"NosyNed" writes:
The phrase "causes Earth to jump from ..." seems to be waving a wand here. If is possible to add just a little explanation? If it is a GR calculation how does the time taken to change direction (and therefor the magnitude of the accelerations) enter into it?
It's entirely handwaving. Essentially a way of putting a balck box around the point of acceleration.
Here is the explanation using the barest hint of General Relatiivty, because I really don't want to use the acceleration metric.
I'm going to go as far as I can with Special Relativity first.
Before acceleration the ship is in frame O' and after acceleration the ship is in frame O''.
We will also say that t(P' or P'') is the age of Earth as viewed from that frame.
The velocities of the frames relative to Earth are v'= 4/5 c and v''= -4/5 c respectively.
However lets see what their velocity are with respect to eachother.
From the relativistic velocity addition law we find this velocity to be 40/41 c.
And the respective gamma factor is g = 41/9.
This means that the time dilation effect between the O' and O'' frames is such that a time interval of t(P') = 9 years in O' is viewed as t(P'') = gt(P') = 41 years in the O'' frame.
The General Relativistic reason for this effect basically comes from the equivalence principle.
So the acceleration a point P from frame O' to O'' is basically the same as an object falling from one height above a giant planet to another height and the effect comes from gravitational time dilation between these two heights.
I'll go into more detail if you want.
"cavediver" writes:
I think the biggest problem with the Twins Paradox is that two very different phenomena are being discussed and confused. We have inertial frame transformation laws and we have the observers' proper time. The real paradox is why does anyone think this is still a good scenario to discuss in the teaching of SR, other than after the teaching of proper time...
What do you think?
I know what you mean. In fact one of my biggest issues when people teach SR is that they downplay or don't deal with proper time.
Which is unusual given its importance in GR.
To be honest I always thought the "pole in the barn" paradox should be introduced first, if you going to bother with paradoxes at all.