And yet you refuse to apply time dilation corrections to him like you did to twin B and arrive at the same answer of 16 years... Funny how your claims contradict your own claims....
Actually, if you looked at what I wrote, I noted that when 8 years had elapsed for twin B (when he turned around),and that 6.4 years had elapsed at that point for twin A.
But, at that point, twin B has to change inertial frames, which means he has to recompute the position of twin A in both space and time. That is what the Lorentz transformation does. He then applies the time dilation appropriate to determine the extra amount of aging twin A goes through, which was 13.6 years, for a total of 20 years. The problem is that when the frames are shifted, the point 'simultaneous' with twin B changing frames, but a distance of 4.8 light years away (.6*8) becomes a point a bit over 10 light years away and a bit less than 9 years in the past. If you then do the time dilation to the time when the twins meet up, you get the extra 13.6 years.
Again, simultaneity is not absolute. When shifting frames, you have to find both the space and time coordinates in the new frame.
from the simple fact you rerfused to apply to A the same calculations you applied to B to arrive at B's age.... Instead you found it necessary to treat A as stationary and apply only time dilation corrections to his frame... I didn't say it, you said it in the very math you used.....
No, I treated A as moving in B's frame(s). In each case, I found the elapsed time in the appropriate frame, the distance gone, and computed the proper time as sqrt(t^2 -x^2), which is the correct formula for proper time (time experiences by the one going on that path). Now, to *find* the appropriate t and x in moving between frames may take some work, but the answer always comes out. Now, if someone is stationary in that frame, then x=0.
No he doesn't, he sees himself at rest. He sees A taking 8 years for each half of the trip. Yet you keep wanting to consider only A as the absolute frame while claiming B's viewpoint is equally valid. Your contradictions are plain to see...
And yet you only apply it to B's frame back to A......
I was computing how much A aged. That requires finding out how far A moved and how long it took to do so *in A's frame*. I did that. For example, when twin B is half-way through the journey, twin A has aged 6.4 years.
And yet you found it necessary to apply the Lorentz transformation which as you put it "tells how to get from the description in one frame to the description in another."
Yes, I was finding what the spacetime coordinates are for the same point in two different frames. If you want to shoft from B'a outgoing frame to his return frame, a Lorentz transformation is required. For A, however, there is no change of inertial frames.
So which is it? Contradictions in every statement you make...
You mean only by knowing what A sees can we then deduce the correct passage of time in A's frame from B's frame. B sees the same exact thing as A. So knowing what B sees, we can determine what A sees, is this not what you just stated? Yet You found it necessary to apply slowing clocks only to B to get 16 years, while refusing to apply the slowing of clocks to A to get 16 years, despite the small fact that this is only what B sees.....
No, B does NOT see the exact same thing as A! Twin B experiences two different inertial frames: an outgoing one and a return one. Twin A, on the other hand, only experiences one inertial frame.
So, yes, if we know what B measures for the position and duration for where A is in space time, then B can correctly determine how much A has aged.
The slowing of clocks is one part of the Lorentz transformation. And yes, if you understood what I did, you would see I *did* apply the time dilation for the time differences in B's frame. That was what the sqrt(t^2 -x^2) is all about, after all.
To paraphrase yourself "How can you say he 'really has'? From what frame do you say that?" B never changes frames from his own viewpoint. A changes them..... Once again only treating A's frame as the absolute frame from which the motion derived....
This is where you are wrong. Twin B *does* change inertial frames. Twin A does not.
You showed nothing except that you had to use A as the abso;lute frame to base all your calculations from....
Once again, any *inertial* frame is equally good. But twin B doesn't stay in a single inertial frame.
To paraphrase yourself "How can you say he 'really has'? From what frame do you say that?" B never changes frames from his own viewpoint. A changes them..... Once again only treating A's frame as the absolute frame from which the motion derived....
No, I even showed that the calculations were for the outgoing frame for B. In *that* frame, twin A has moved 15 light years from the beginning of the whole scenario to the end. Also, the whole scenario too 25 years in that outgoing frame. And the time dilation applied to that 25 years is .8*25=20 years.
But, twin B does *stay* in that inertial frame. Twin B *changes* frames half way through. So, you have to compute the change from the outgoing inertial frame to the return inertial frame, both of which are moving with respect to twin A (at 60% of c), but also moving with respect to each other(at 88.2% of c).
Yes, you could take B's viewpoint that A's clocks are slowing and calculate 16 years just like A did, but you won't.... because you'll still consider A as the absolute frame....
I won't because that 16 years is not the time from any single inertial frame. It is a mixture of times from two different frames. That means you have to compute how to shift from one to the other.
Pseudoscience. The Hafele–Keating experiment did all calculations within the same earth centered frame, despite direction changing..... The slowing had nothing at all to do with magical pseudoscientific frame switching.... Of course this experiment was not performed until after the pseudoscience of frame switching was set into literature as fact. But why let reality get in the way of a good story, right????
Yes, all are related to the frame of the Earth. We can use any inertial frame to do the required calculations and get the same answer.
But once again: To paraphrase yourself "How can you say he 'really has'? From what frame do you say that?" B never changes frames from his own viewpoint. A changes them..... Once again only treating A's frame as the absolute frame from which the motion derived....
OK, so you don't grasp the notion of an inertial frame. Twin B changes inertial frames and twin A does not. it really is that simple. Twin B will experience an acceleration because of that change while twin A will not. In a spacetime diagram, the path of twin B will change slopes while the graph of twin A will not. And these statements will be true for *every* inertial frame that looks at the situation. That is the *basic* asymmetry between the two twins: one experiences an acceleration, while the other does not. The reason is that the path of one (twin A) is a straight line in spacetime while the path of the other (twin B) is NOT a straight line in spacetime.
To paraphrase yourself "How can you say he 'really has'? From what frame do you say that?" B never changes frames from his own viewpoint. A changes them..... Once again only treating A's frame as the absolute frame from which the motion derived....
Let's put it this way. Look at twin B's outgoing frame. Twin B sees that frame as at rest while be is in it. But, from twin B's outgoing frame, the return frame is going at 88.2% of c. Twin B has to change from the frame where he started out seeing himself being at rest to a *different* frame that moves at 88.2% of c with respect tot hat first frame.
Conversely, from B's return frame, the outgoing frame is moving at 88.2% of c. This has NOTHING to do with twin A. it is simply a matter of twin B changing from one frame to another frame at the middle of journey.
If you see yourself at rest at one moment and then experience an acceleration, you have changed frames even if you see yourself as at rest after that acceleration. To be in the same frame means to always be moving at a uniform velocity in the same direction and thereby not ever experiencing an acceleration. That is the *definition* of an inertial frame: one which does not experience an acceleration 'force'.
Ahh, so reality only intrudes into your consciousness when it is B's actual motion, but then we can't say A isn't stationary because then from what frame would you say that. lol, you people are a riot....
Says who? Not B..... B says he stays in the same inertial frame the whole time and A does not. Please decide whether you believe A is the absolute frame or not....
NO, once again, twin B does NOT stay in the same inertial frame.
To paraphrase yourself "How can you say he 'really has'? From what frame do you say that?" B never changes frames from his own viewpoint. A changes them..... Once again only treating A's frame as the absolute frame from which the motion derived....
No, twin B changes from one frame where he was at rest to another frame moving at 88.2% of c with respect to the first.
Both versions treated A's frame as the absolute frame and only B in motion. B does not see this...
No, they do not. But they *do* take into consideration that B changes frames.
And you would be wrong except done from Frame A.... As shown by the fact you don't really want to consider B's viewpoint as equally valid.
If twin B stayed in the same frame the whole time, there would be no issue. But twin B does NOT stay in the same inertial frame the whole time. That is the whole point.
We could take the reference frame for everything to be twin B's outgoing frame. Do you want me to do all the calculations in that frame?
OK, twin B starts out in the outgoing frame. In that frame twin A is moving at 60% of c. After 8 years in that frame, twin A is .6*8=4.8 light years away and has experienced .8*8=6.4 years (time dilation here).
Now, twin B starts to move with respect to the outgoing frame at a speed of (.6+.6)/(1+.6*.6) =.882= 88.2% of c to catch up with twin A. This is motion with respect to the outgoing frame for B. It takes (distance divided by difference in speeds) 4.8/(.882-.6)=17.0 years to catch up. Now, take time dilation into account again, and find that twin A has aged an additional 17.0*.8=13.6 years while twin B was catching up. The total amount of aging for twin A is then 6.4+13.6=20 years.
Now, of course, twin B also ages during that time. In the outgoing frame, it takes 17 years to catch up, so applying time dilation again, twin B ages an additional 17*sqrt(1-.882^2)=8 years. Hence, twin B experiences a total of 8+8=16 years for the whole scenario.
Once again, you have to stay in the same inertial frame if you want to do the calculations correctly. There are three inertial frames in this scenario: the frame that twin A stays in the whole time, the outgoing frame for B, and the return frame for B.
