In order to understand why the twin paradox goes away, I first have to highlight a point of confusion with Einstein’s transformation equations. In the model of Complete and Incomplete Coordinate Systems, there are four time equations; one for the time to travel along each of the 3 axes, and one for the amount of time that the coordinate system has been in motion.
In an Incomplete Coordinate System, time may appear to run slower. But this “appearance” of running slowing is completely dependent upon how we determine to measure time. Consider our example with the birds in a moving cage where the bird is flying at constant velocity wfrom left to right and back again. We can decide to measure time by counting each time the bird comes to the left side of the cage. This represents one unit of time. When the cage moves forward with velocity v, the time lengthens. This makes since as the bird has to fly further. The amount of time now needed for the bird to fly this distance is found by multiplying t by 1/sqrt(1-v^2/w^2).
Some will point to this example and say, “a-ha, this is Einstein’s equation for time dilation.” Now add the cat to the cage. When the truck is stationary, the cat travels at a constant velocity w from the left to the right and back again. Now we decide to measure time by counting each time the cat comes to the left side of the cage. When the truck is stationary, the cat and the bird return at the same time as they both travel the same distance.
However, when the truck is moving forward at velocity v, the bird travels through the air, behaving as an Incomplete Coordinate System. The cat, however, walks on the floor of the cage, behaving as a Complete Coordinate System. In this case, the cat’s time does not elongate as the bird’s did as the cat doesn’t have to walk any further than he did when the truck was stationary.
As you can see, if you count time using the oscillations of the cat you will have a different result than if you count oscillations of the bird. So, if we’re measuring time using the cat, time remains the same. If we use the bird, time elongates. Are the cat and the bird experiencing different times in the same coordinate system? Of course not.
What does it mean?
>In our example, if you chose to measure time using the oscillations of the bird, then time would seem to elongate. If, however, you use the cat, or even something external to the cage (say the watch on the truck driver’s arm), then time would not elongate.
We need to understand that there are four time equations instead of one. Three of them indicate how long it takes to travel certain distances along the X, Y, and Z axes. The fourth indicates how long the coordinate system has been in motion. It is important that these not be confuse with one another. Einstein only recognized one time equation which contributes to the confusion. It is this confusion in the understanding of the time equations that results in the misinterpretation of time dilation.