Re: Ralfativity vs Relativity: synchronicity
Well I've figured it out. Relativity should have been named the theory of synchronicity. It's like Bob and Alice are mirror images of each other except both can control the others image. A change in one will cause a change in the other. The farther apart they are from each other, the more lag they'll see between their change and the change in the image. The lag is where relative aging comes in until the images resync.
Relativity, on the other hand, has been the study of mirages; time dilation, length contraction, relativistic mass and relativity of simultaneity, all illusions. But the chief "mirageologist" of relativity was quite skilled. He was able to conjure up realities from the illusions: relative aging, the conversion of mass/energy, the nature of gravity, just to name a few. For me, the realities are what's important, not the illusions.
Synchronicity concerns itself with mirrors, not mirages. For relative aging, the mirror is the line of velocity. As soon as there's a change in the velocity of one there's a delay in the change of relative velocity between the two. Until the images re-sync, relative aging occurs.
So in my previous example, when tb = 3 and t'tx = 1.5, 1 and .75 (1/r=2,3,4) for .6c, .8c and .8824c respectively, synchronicity demands t'a = 3 for ttx =1.5, 1 and .75 (1/r=2,3,4) for .6c, .8c and .8824c respectively. Note ttx is the light line from Bob's velocity line (the vertical ct axis) to Alice's velocity line while t'tx is the light line from Alice's velocity line to Bob's. Perfect symmetry reigns until a change occurs and there's a speed mismatch for a while. The ratio of r's is always 1 before a transition but the ratio of r's is not 1 during the transition period and then returns to 1 after. The mirror gets fractured by the transition period so the light lines get shifted. I need to go through an example to show what this means as it's new and quite complex.
For example, Alice leaves at .6c and returns after 4 of her years at .6c. We know the transition period lasts from t'p = tb = 4 (they age the same before the transition) to tb = t'pY1*(c-v1)/c =8. So Bob ages 4 yrs during the transition. Now we need to figure out how Alice's relative aging unfurls during the transition per each of Bob's 4 years in real time.
Although it looks like Bob's aging at the same constant rate, what causes his aging during his 1st 4 years is different from what causes his aging during the 4 yr transition period. For the 1st 4, tb = t'a up until t'a = t'p. The derivation of Alice's age from Bob's age is the same for each year. We've already done it for tb =3 but now we'll do it for tb=4 which is right at the transition point.
At .6c and tb=4, t'tx =2. The mirror move happens when you make ttx = t'tx and derive the corresponding t'a using the formula t'a = cttx*Y1(c-v1). So if tb =4, t'a =2*1.25(1.6) =4. The mirror also works in reverse for finding tb from t'a.
The next 4 years for Bob are derived from the remaining 2 years for Alice before the transition. For Alice 2 to 4, Bob ages 2 years for each year Alice ages. Let's go through the analysis year by year to understand how the mirror is fractured by the transition. The fracture forces a shift of the results of t'a for tb=2 to 6 to tb=4 to 8.
t'a =4.5 ttx = 3 but ttx can't be equal to or below tb=4 so it must be shifted to tb = 5.
t'a =5 ttx = 4 but ttx can't be equal to or below tb=4 so it must be shifted to tb = 6.
t'a =5.5 ttx = 5 the shift continues so add 2 to 5 and you get tb =7.
t'a =6 ttx = 6 the shift continues so add 2 to 6 and you get tb =8.
tb=8 is the upper bound of this transition period as calculated before. So for this period, Bob ages 4 years while Alice ages 2. You can see for each year Bob ages, Alice ages half a year. That is how the universal present is decided, not by the formula t=t'Y as it is for relativity.
Luckily I finished on time today. I'll go through another more complex example tomorrow which will help me derive a formula that governs this nature of relative aging.