My new simultaneity method is an alternative to the commonly used co-moving-inertial frames (CMIF) simultaneity method. I personally prefer the CMIF method to my method, but some people dislike the CMIF method because it produces an instantaneously changing current age of the home twin (she), according to the traveling twin (he), when he instantaneously changes his velocity. That instantaneous changing of her age, according to him, is especially abhorrent to some people when it is negative, i.e., when she instantaneously gets YOUNGER, according to him. I personally am not bothered by that (and neither is the well-known physicist Brian Greene, as was shown in his PBS Nova series on "The Fabric of the Cosmos"). My simultaneity method provides an alternative "safe haven" for those people, because it has no discontinuities (either positive or negative) in her age, according to him.
My simultaneity method is described in detail in a monograph published by Amazon, available for $5.00. You can find it most easily by searching on my complete name: "Michael Leon Fontenot" (with the quotes).
That monograph addresses only the issue of how to determine her age according to him, versus his age. I.e., it only addresses simultaneity. But I have also recently worked out what my method has to say about how he can determine their distance apart, according to him, at any instant in his life. That addition makes my method a complete coordinate system for him. Here is a description of how my method determines their separation, according to him.
The CMIF method and my method give the same answer to the question "What is the distance between the home twin (she) and the traveling twin (he), according to him" ONLY when there is no discontinuity in the CMIF solution. There is no discontinuity in their separation in the most common version of the twin paradox, where he reverses course at the turnaround point, but comes back at the same speed as he used on the outbound leg.
But if he uses a DIFFERENT speed on the return leg than he used on the outbound leg, the CMIF solution gives a discontinuity in their separation when the speed changes (according to him). My method never produces a discontinuity.
For example, suppose his speed on the outbound leg is 0.57735 ly/y (gamma = 1.2247). (That case is advantageous, because the angle of his worldline wrt her worldline (the horizontal axis) is 30 degrees, and can be easily drawn with a 30-60-90 plastic triangle.) Suppose she is 40 years old when he turns around. He is 32.66 years old then. Then suppose his new speed is -0.866 ly/y (gamma = 2.0). When they are reunited, she is 66.67 years old, and he is 46.0 years old.
For the above case, we can plot their separation at each instant of his life, according to him, for both the CMIF method and for my method. I call that diagram the "SAAOD", for "Separation According to the Accelerated Observer Diagram." Their separation is on the vertical axis, and his age is on the horizontal axis.
The first straight line segment starts at the origin, when he is zero years old, and their separation is zero ly. It slopes upward at a 30 degree angle. It reaches its peak when he is 32.66 years old, and their separation is 18.86 ly.
When he changes his speed to -0.866 ly/y, their separation instantaneously drops to 11.55 ly, according to the CMIF method. Then, from there, their separation declines linearly with a slope -0.866 until their reunion. (That last straight line segment makes an angle of approximately 41 degrees wrt the horizontal axis).
In contrast, in my method their separation doesn't instantaneously change. Instead, it decreases linearly with a slope of -2.047 from its peak at 18.86 until his age reaches 38.85 years old, and their distance reaches 6.188. (That endpoint corresponds to when he receives a light pulse that she sends at his turnaround). Then, at that point, the slope of the line decreases to -0.866 until their reunion. The two methods coincide along that last segment.
