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If there is only two inertial frames in empty space with no at rest frame they can plot their trajectories from, how can the two frames determine their worldlines, coordinate trajectories, point of departure and arrival, proper time and distance, issues of time dilation, in the emptiness of space?

If that is too many questions please consider just the first one.

Thank you very much.

(I realize in the real universe this set up does not exist as there is always some type of at rest frame they can use to calculate everything with respect to. I have looked closely to see if this question has been asked elsehwere in this forum or anywhere else, I didn't see it unless I missed it)

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    $\begingroup$ In spite of how things may have been described in school, I think it’s a conceptual mistake to think that a coordinate system must be laid down before geometric figures can be described. We can draw figures and do geometry before assigning coordinates to anything. A similar situation happens in spacetime. To answer your question, you can use Bondi’s radar method. $\endgroup$
    – robphy
    Commented Oct 3, 2022 at 2:26

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Be it in empty space or in presence of matter: There is no absolute way to define a frame as being at rest. Typically you would consider a frame at rest when it is at rest relative to you, the observer. Or, equivalently, you require that the observer is at rest in the reference frame you call at rest.

Btw. "if there is only two inertial frames in empty space..." sounds like reference frames are objects that you find somewhere, like in nature, or on an excercise sheet. They're not. You make a choice of reference frame for the description of a problem, and there are infinitely many such choices. It's only that some are better suited than others... :)

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You could adopt a coordinate system in the inertial frame of one of the observers and calculate everything you want in terms of that reference frame.

In particular, there is no "absolute" time dilation, it is a relative effect seen between observers in different reference frames.

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