You like kilts, I got kilts. Not plaid, more argyle than plaid. I got your Epstink, I got your Loedel right here.
I've redrawn the original Minkowski from Bob's perspective into the new Epstink STD depiction. I've rounded out the numbers and added a few lines. I've also included the Loedel equivalent so you can see how the lines deform from one portrait to the other.
Now everything should be as clear as a ball of yarn. Let's start with the blue gash from 5 to 3. It represents Bob's age as 5 but Alice's age is a combination of her time through time (t') and here time through space (x/c). The formula, according to trigonometry, for this is
t
^{2}=t'
^{2}+(x/c)
^{2}When Bob is 5, the blue line points to Alice t'=3 and x=4 which means Alice's age is 5 from her 2 components of time.
The purple age lines illustrate their same age in the instantaneous present for both within their own frames. Notice how they are the same slope as the lines of simultaneity from the intermediate half speed green frame's perspective. For the Epstink, .5c is half speed and for the Loedel it's 0c. The intersections between the purple and green lines are the same for both depictions (5.7 and 3.4). This illustrates that Alice and Bob are not the same age from either's perspective in the delayed present but they are always the same age simultaneously from the half speed intermediate frame's perspective. (Half speed refers to the velocity combination law from relativity.)
The difference between time dilation (blue line going to a point on the thin red line) and time contraction (red line coming from that same point and being dilated back onto Bob's timeline) should be now graphically apparent. This is what I meant by time contraction is Bob seeing his own time dilated to Alice and then being dilated once again through Alice's perspective. Time contraction is Bob's time being double dilated and physically represents the time Alice takes in her years to traverse Bob's proper distance in Bob's proper time. This construct completely replaces the need for length contraction which will no longer be depicted in the Epstink STD.
So in this example, Alice will traverse 4 ly in 1.8 of Bob's time contracted years which gives a v'=Yv= 1.3333c. Another way to see it is t=Yt' = 5/3*1.8 =3. v'=4/3=1.3333c. Alice crosses 4ly in 1.8yrs mixed perspective v' which is not v so no c speed limit is broken. The distance between 4 light year markers is not contracted, the time it takes to cross those 4 markers is contracted from Alice's time perspective of Bob's proper distance.
In the next post we'll talk about how the yellow and pink light signal lines work to post-process the delayed present to verify Bob and Alice were the same age in the instantaneous present of the past or pre-process that they will be the same age in the instantaneous present of the future. As I mentioned, the light lines are no longer at a 45
^{o} angle for Epstinks and the time they actually travel must be corrected from what is depicted. The bells should now start ringing in someone's head reading this.
PS. I'm thinking of renaming Epstink to the Ralf STD. I mean why should either of them get any credit; they didn't think of it.