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In more complicated scenarios, we can still determine what their two conclusions are about their current ages, assuming that the simultaneity method we choose to use is correct. If the home twin remains perpetually inertial, we DO know that her conclusion about the current age of the traveler is correct ... she can continue to use the more general time dilation equation, which says that the traveler's rate of ageing will be slower than hers, by the gamma factor. But he can't use the time dilation equation. He has to choose one of the four simultaneity methods for accelerated observers, and each method gives a different answer. He might decide to reject two of them (Dolbe and Gull's "Radar method", and Minguizzi's "fictitious twin" method) because they are both non-causal. That leaves the CMIF (Co-Moving Inertial Frames) simultaneity method, or else my simultaneity method. I prefer the CMIF method, but some people dislike it because it says that in some circumstances, the home twin can get younger, according to the traveling twin. Those people may prefer my method, which has no discontinuities and no negative ageing of the home twin. Which of them (CMIF or mine) is correct (if either) is unknown. I believe that there IS a correct simultaneity method (for philosophical reasons), but I don't know what it is.

I've worked out an example that shows an instantaneous negative ageing for her, according to him, when CMIF simultaneity is used ...i.e., she instantaneously gets YOUNGER according to him, when using the CMIF simultaneity method. And I'll show the results for this example for my simultaneity method, which gives no discontinuities and gives no negative ageing.

First, here's a description of the Minkowski diagram (with time tau on the horizontal axis and separation X on the vertical axis). That always has to be determined first.

The two "twins" in this case aren't really twins ... they are just babies who were born at the same time but 20 lightyears (ly) apart, and with zero relative velocity. This situation continues until they are both 40 years old. We represent this initial situation by drawing a horizontal line on the diagram at the point X = 20 on the vertical axis, extending from tau = 0 to tau = 40. That is the initial segment of his worldline. At the end of that first segment, write 40 immediately above the end of that segment of his worldline to show his age , and vertically below there write 40 immediately below the horizontal axis to show her age.

Then, he instantaneously changes their relative velocity to v = 0.57735 ly/y, and continues that velocity for the rest of their lives. This causes his worldline to slope upward toward the right at a slope of 0.57735 (and an angle wrt the horizontal axis of 30 degrees). Label that point where the second segment of his worldline starts as point T.

Next, draw a 45 degree line starting at the point 40 on the horizontal axis, and sloping upward to the right, representing the worldline of a light pulse that she transmits when she is 40, and that is moving toward him. We then write an equation giving X as a function of tau for that light pulse, and then we write another equation giving X as a function of tau for the upward sloping segment of his worldline. Then, we set those two equations equal (force their X values to be equal). The result gives the value of tau where those two lines intersect ... label that point Q. That point is vertically above the point tau = 87.32 on the horizontal axis. Write that value just below the horizontal axis, vertically below that point of intersection.

We also need to determine their separation according to her (the value of X) when she is 87.32. The answer is 47.32 ly.

Next, we need to plot two lines of simultaneity (LOS's) that show what "Now" is for him. (The LOS's for her are just vertical lines). His LOS's (anywhere for him when his velocity is 0.57735) have slope 1/v = 1/o.57735 = 1.73, and they make an angle of 60 degrees wrt the horizontal axis. The first LOS we need goes through point Q. That line intersects the horizontal axis at the point tau = 60. That is determined by writing the X(tau) equation for that LOS, and solving it when X is set to zero. So this tells us that when he is 78.63 years old, she is 60, according to him.

Next, we need to determine how old she says he is when she is 60. We know that according to her, he ages slower than she does by the factor gamma = 1.2247 (once he has changed his velocity to 0.57735). So according to her, while she ages from 40 to 60, he ages from 40 to 56.33. Mark that age on his worldline.

We also need to determine their separation according to her (the value of X) when she is 60. The answer is 31.55 ly.

Next, we do the same thing for the LOS that goes through the point T where his worldline starts sloping upward. The result is that her age when he changes velocity is 28.45, according to him. He was 40 then.

From the above information, we can draw the Age Correspondence Diagram (the ACD), which is a plot of her age (on the vertical axis), according to him, versus his age (on the horizontal axis).

During the first segment, their relative velocity is zero, so they each agree that they are ageing at the same rate. Therefore the first segment of the ACD is just a line of slope 1, sloping upward to the right, making a 45 degree angle wrt the horizontal axis. This first segment is the same, regardless of whether you are using the CMIF simultaneity method, or my method. Label the end of that segment point T.

In the CMIF method, at point T, when he changes velocity from zero to 0.57735, he says that she instantaneously gets younger by 11.55 years, from 40 to 28.45. So, for the CMIF case, we draw a vertical line downward from point T, of length 11.55 ly. Then, the next (last) segment slopes upward forever at a slope of 1/gamma = 0.8165.

What does the plot look like after the point T in the case of my simultaneity method? It is a straight line between the point T and the point Q. Point Q is where his age is 78.64 and her age is 60. It is a point on the third segment of the CMIF line we determined above. Point Q is where he received the pulse from her, and it is the end of the "Disagreement Interval" (DI) between him and the a perpetually-inertial observer who is co-located and co-moving with him. So, after point Q, the ACD for my method coincides with the CMIF method for the rest of their lives. I.e., after the end of the disagreement interval (DI), the CMIF method and my method agree thereafter in this example.

So, as was claimed, with my method, the ACD has no discontinuities, and no negative ageing (i.e., the ACD plot never slopes downward).

I personally prefer the CMIF method, because of its simplicity, and because I'm not bothered by discontinuities or by negative ageing. But for those people who ARE bothered by those characteristics of the CMIF method (and in my experience, that's a LOT of people), my method offers a safe refuge.