Re: Ralfativity vs Relativity: post transition analysis.
We're going to consider several scenarios for what happens after Alice turns 4: Alice returns to earth at light speed, Alice returns at .6c, Alice stops wrt Bob, Alice stops wrt Bob instantaneously and continues away at .6c, Alice moves away at light speed. Hopefully as I consider all perspectives I'll see some mathematical pattern that will lead to a rule where I'll no longer have to consider all perspectives to find 2 answers that match.
1. Alice returns to earth at light speed after she ages 4 yrs her time.
This is not possible but it does reveal some interesting insights as a limiting case. Actually I've read ahead and this example reveals everything about the true nature of relativity and it's very exciting.
First, it will take 3 years for her to return to Bob because she can't traverse that distance faster than light. She won't age a second in that time; she'll view the journey as instantaneous. Her TV picture, when she was 3ly away, will hit Bob at the same instant as her arrival. Bob will be receiving that frozen picture at hyperfast speed so it will still look like she is not moving. The TV picture up until that point was of Alice going at half speed.
The wavelength of the light signal carrying the frozen/hyperfast TV picture is blue shifted to the max by the classical Doppler effect but its wavelength is not frozen in time just because light travels at the speed of light. If it was then we wouldn't be able to see any light frequencies because they would be frozen in time. Luckily light has no velocity through time, since all its velocity is through space, so its time relative to ours flows at the same proper time as ours.
According to ralfativity, Y=1 at v=c which reduces Alice's velocity through time from infinity to zero and she's only left with the speed of light through space. At 0 speed through time, time is at a standstill for her but she travels 3 years through space at light speed to reach Bob. That trip ages Bob 3 years. But hold on a tick, didn't we just agree that at the turnaround point, Bob and Alice had both aged 4 years? 3+4=7 yrs for Bob yet the STD says Alice meets Bob when he has aged 8 yrs and she has aged 0 years in getting back to Bob. What gives here? Pure beauty that's what.
In ralfativity each time unit on Alice's velocity is t'=t/Y. 1/slope of Alice's line is therefore v using Bob time or Yv using Alice time. With Y=1 when v=c, The first year of her journey at light speed back to Bob, Alice will receive a signal from Bob that he is 4. Just before her turnaround, she had received a signal from Bob that he was 2 yrs. In the 2nd year of Alice's trip back to Bob at light speed, she receives a signal from Bob that he has aged 6 yrs. So for every year at light speed Alice travels back to Bob, he ages 2 years from the time he was 2 for a total of 8 yrs. Alice would see Bob's TV picture at double speed after the turnaroundif the journey for her wasn't actually instantaneous. I know, confusing things happen at light speed.
She would see the same effect on Bob's TV picture speed if she had returned at .6c but she would see double speed for Bob's years from 2 until 10 instead of ending at 8. For a return speed of greater than .6c, Bob's picture speed will approach infinitely fast and will only look infinitely fast at v=c to Alice because, to her, the trip to Bob is instantaneous.
This is in total agreement with what ralfativity predicts from the STD and in total disagreement to relativity's incorrect interpretation of the STD using reciprocal time dilation. However, if relativity replaces reciprocal time dilation with the doppler shift timing analysis, relativity would be in agreement with ralfativity (except for the conclusion that Y=1 at v=c).
2. Alice accelerates away from earth to c after 4 yrs.
This is also not possible but it illustrates how different the results are from turning around at light speed. Alice will see Bob frozen in time at 2 because she will miss his subsequent transmissions. For every ly she travels, she won't age but Bob will age 2 yrs. So if she goes out a million light years, he will age 2 million years and she won't age at all. The limit on his age will be the size of the universe but is not infinite.
3. Alice continues away from earth at .6c.
She and Bob will continue to see each others TV picture moving at half speed. The timing analysis will be the same as that for pre-turnaround, Alice's age will remain the same as Bob's and flow at the same proper rate of time as his. The reciprocal time dilation between them will just be an illusion of perspective and irrelevant to reality as it has no lasting or meaningful effects.
4. Alice instantaneously stops and restarts her travel away from earth at .6c
This example illustrates that reciprocal time dilation is not masking relative aging and that acceleration does not magically unveil relative aging that was occurring all along for the past 4 years. In fact a quick stop and restart will not change the fact that Alice and Bob will continue aging at the same rate as can be seen from the STD. There will be no instantaneous discontinuity of age between the two after the stop/restart. The mechanism of how a stop affects relative aging will be seen in the next example (very important).
5. Alice stops wrt Bob.
This scenario is basically a transition between no relative aging (both 4) at a non-zero relative velocity (.6c) to no relative aging (at some point after the transition they both age at the same rate again) at zero relative velocity resulting in a permanent difference in aging (1yr for our example). We know before the transition Bob and Alice both aged 4 years. Relativity, as taught to me, says right at the stop point, Alice can immediately calculate Bob's age instantaneously changes from 4 to 5. (Of course this was proven false in example 4 above.) Bob will verify this fact 3 yrs later when Alice's result reaches him.
This fact cannot be altered by whether you consider Alice from the moving or stationary perspective as the perspective does not alter who has initiated the stop. This is not the same thing as either Alice or Bob initiating the stop. If Alice is deemed moving and initiates the stop, then she can immediately calculate Bob instantaneously ages from 4 to 5 as soon as Alice stops (an impossible conclusion). Maintaining the same perspective, if stationary Bob jumps to .6c thereby achieving 0 relative velocity to Alice, then he can immediately calculate Alice has instantaneously aged from 4 to 5.
Lets employ ralfativity timing analysis to see what's really going on after the transition point when both Bob and Alice are 4.
1. Alice moving sends out her 4 yr mark to Bob who receives it 3yrs later when he is 8. So we need to calculate Alice's age when Bob receives her message assuming nothing changes. Add 3 to Alice's age and she is 7 when Bob is 8. She has aged 1 yr less than Bob and that will remain the same as both age the same subsequently. But lets work backward to see how this relative aging has unfolded.
Alice moving sends out her 3yr mark to Bob who receives it 2.25yrs later when he is 6. In .75 yrs of light travel, Alice reaches 4 which leaves 1.5 yrs of light travel after the transition so she will be 5.5 when Bob is 6. So when Bob's age moves from 6 to 8, Alice has moved from aging a half year less than Bob to a full year less than Bob as calculated by Bob.
Alice moving sends out her 2 yr mark to Bob who receives it 1.5 yrs later when he is 4. We've done this before and we know Alice is 4 because they're the same age pre-transition. So Bob can calculate upon receiving the news of Alice stopping (Alice's TV signal will transition from being seen at half speed to normal speed), that during the previous 4 years Alice has been aging only .75 yr for every year Bob ages until he reaches 8. After that, Alice will age 1 yr for every year Bob ages. As you can see, Alice ages slowly (only 3 yrs) over the 4 yrs after the transition. The change is not instantaneous nor does it begin until the stop nor is it a conversion of accumulated time dilation from the pre-transition 4 yrs. It's more like the unit size of Alice's clock ticks is compressed by the stop and the excess time has to be bled off at a precise rate so the relative time rate between the clocks can achieve a new harmonized proper time rate.
That's all for today. I need to continue this analysis from all perspectives to figure out a way to recognize which 2 analyses yield the same correct answers. I also need to fully analyze the scenario when Alice returns to Bob at .6c.