Einstein used light signals to synchronize clocks in the same reference frame to the same time no matter the distance separation between those clocks to come up with some kind of perspective "present". But can the opposite be done? Can you use light signals from clocks in different reference frames to calculate what your proper time "present" was when your twin-paradox twin sent you a light signal with his proper time indicating the moment he sent it?
Below is a Minkowski diagram of a twin paradox example for Alice doing a round trip at 0.6c. Bob sends out a pink light signal to her when his proper time is 2. She receives it when her proper time was 4. She wants to figure out what was her proper time when he sent it.
From her perspective, Bob's light signal is sent when she is 2.5 and Bob's 3 (Bob) yr light signal was only 1.5 (Alice) years long from the time she's 2.5 until she receives it at 4. So her proper time was 4 - 1.5 (light travel time) - 0.5 (the relativity of simultaneity from when Bob actually sent the signal to when she saw him send it) = 2. Her proper time was 2 when Bob sent the signal so we draw a green proper time line of simultaneity to signify the answer.
Now I can figure out the rest of the answers for each light signal from Bob and Alice but my math method is not appreciated here. Does relativity have a method to calculate proper time simultaneity using light signals from each year in the diagram? You don't even need to do every year just from Bob = 5. The answer I get is Alice's proper time was 4.5 when he sent his light signal when his proper time was 5. The reciprocal also seems to be true, that if Alice sends out a signal at 4.5, Bob's proper time was 5 when she sent it.
For bonus points figure out the light travel time from Alice's perspective. The remainder should be her relativity of simultaneity of when she sees Bob release the light signal. I'm trying to figure that out myself right now.
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