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This is article about Bondi K-Calculus.

https://en.wikipedia.org/wiki/Bondi_k-calculus#cite_note-Bondi80-7

What I learned from that article (for example when determining who was old in twin paradox) was when staying twin determining what time in traveling twin frame that simultaneous to him, he used radar methodology, but when traveling twin determining what time in staying twin that simultaneous to him, he used angle between coordinate. For me it is not fair, why traveling twin did not use radar methodology too?

Second, if we switch between who was sending the signal with who was replying, then the situation become alter too. So this method does not make a symmetrical treatment to explain time dilation. If traveling twin sending a signal, then staying twin that younger.

Third, kT was the time for traveling twin when he receiving the signal and then send replying signal. But why this kT become time for traveling twin to reach that point (spatial location when he replying the signal) and it is attributed to time dilation effect. I think it is just because the signal itself have a delayed time to reach traveling twin, not because time in his reference frame that dilated, isn't?

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I assume that these questions deal specifically with the [simple] twin paradox, where the twins reunite after separation with one twin inertial and the other non-inertial [but piecewise-inertial].

By the time the twin paradox is examined, it is usually presumed that an eternally inertial observer has a well-established method for assigning coordinates for all of Minkowski spacetime. While another eternally inertial observer might assign a different set of coordinates, both of these observers will agree on the magnitudes of all worldline segments.

  • Concerning Q1 and Q2:
    In the twin paradox, since the "traveler" is non-inertial, he is in a different category of observers... though he may go through all of the same methods to assign coordinates as an inertial observer, his assignment is not easily transformable (by a Lorentz transformation) to the assignment by an inertial observer. It might even be suspect (e.g. if an event is assigned two sets of coordinates). Thus, there is [or there should be] no expectation of "fairness".

    One could ask "what the non-inertial observer would measure"... but that would certainly depend on the details of the measurement procedure (radar vs clocks-and-metersticks vs other methods...). That's another question altogether... and not an easy question, especially without details. [See, for example, "On Radar Time and the Twin `Paradox'" by Carl E. Dolby, Stephen F. Gull - https://arxiv.org/abs/gr-qc/0104077] So, that aspect isn't usually discussed in an introductory presentation. Thus, the determination of coordinates and segment-magnitudes by the inertial observer is good enough to discuss the effect that the traveler's elapsed time is shorter than the home-twin.

    Note: from separation to just-before the traveler turns around, the traveler is inertial (and is thus equivalent to the home-twin).. but once the turn around happens (which can be revealed by a ball on a frictionless table in each frame) the traveler is no longer inertial.

  • Concerning Q3:
    The assignment of $kT$ to the traveler's reception event is
    first about geometry [without any specification of the value or meaning of $k$, other than it is a constant when dealing with inertial observers],
    then later it gets its physical interpretation as the Doppler factor (which incorporates time dilation in Special Relativity).

    Geometrically, it just says, if the home-twin waited until 1.5T to send the radar signal, the traveler would receive at elapsed time proportional to 1.5T--- call it $k(1.5T)$ with proportionality constant $k$... and at this point, nothing else is known about $k$ other than it is greater than 1 for separating inertial observers. As similar construction would apply for sending a sound wave in Galilean relativity (where this is no time-dilation). When the rest of the radar method is implemented and interpreted, then we find the relation of $k$ to the relative velocity.

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