Emboldened by Peter's essay on frames versus coordinates, I thought of producing a little essay on the relationship between my post #31 and Peter's #35 (along the way making terminology consistent between these posts). However, I think the a more important concept to address is the pervasive (and false) idea that it is reasonable for a non-inertial observer to assume distant positions and times correspond to those of an inertial observer momentarily co-located and stationary with respect to the non-inertial observer. This gets at some of the same issues as my prior plan, but with a more physical slant.
The first thing to ask is how does an inertial observer get from a local frame as described in #35 to a global, natural, coordinate system (which, in SR, is often called - sloppily - a global inertial frame, rather than an inertial coordinate system)? There are actually several ways, which all happen to produce the same result for the inertial observer. Simultaneity can be established with Einstein's simultaneity procedure. Distances can either be measured by two way light speed or rulers matched to a standard ruler, or by parallax. For inertial observers, it doesn't matter what (reasonable) procedure you choose, the result will be the same. One way to conceptualize the resulting coordinates is a grid of clocks connected by stiff springs that are never under tension or compression (in a moment it will be clear why specify this rather than rulers). The clocks, once synchronized, keep mutually consistent time (comparing them any time later shows they are in synch, accounting for light delays). Obviously, you don't actually construct such a thing, but you certainly would think reasonable coordinates would be equivalent to such a thing. A fundamental point is that the equivalence of different procedures produces a unique natural choice.
Then, you should ask, what are reasonable (global) coordinates for a non-inertial observer? The surprising answer (expanded on below), is that there aren't any! A little more precisely, there are many possible choices, but none of them satisfy the expectations you might have from the inertial case. Further, each fails in different ways, so there is none to prefer. As a result, there is simply no fixed meaning to be attached to a statement like "how far away is that distant object" for a non-inertial observe. The best you can do is say: "choosing to use procedure x (rather than y), the distance is blah". Even more surprisingly, while available procedures include radar ranging and parallax, one that is not available for all non-inertial observers is rulers!
Ok, so let's try by build physically motivated coordinates for a non-inertial observer. One might assume we can posit a grid of clocks connected by springs such that the springs don't stretch or compress. We allow that each clock is independently propelled as needed, to achieve this. We don't even worry, for now, about synchronizing the clocks. The question is, can we set up such a grid for a rocket that starts at rest and accelerates [edit: and turns], then coasts. The surprising answer is no, this is impossible, in principle in SR! The relevant theorem is Herglotz-Noether. Thus, the most basic concept of what a coordinate grid means is unachievable for a [edit: some ] non-inertial observer. There are different ways of understanding this, but one way is that trying to apply the constraint that the springs don't stretch or compress requires that some clocks move super-luminally (or worse, backwards in time). This same theorem implies that the concept of a long rigid ruler for a [edit: general] non-inertial observer does not exist, even conceptually.
On recognizing that it is impossible to build rigid grid coordinates for a non-inertial observer, what can you do? The two most common choices are radar coodinates or Fermi-Normal coordinates (developed for GR but useful for non-inertial observers in SR). At laboratory scales (or within a rocket), they are identical to plausible experimental precision. Radar coordinates are operationally defined and can be extended globally as long as the defining observer's world line is inertial before some event, and inertial after some event. The beginning and end inertial motions need not be the same. Fermi-Normal coordinates try to glue together momentarily co-moving inertial coordinates, to the extent that it is possible. As implied by Herglotz-Noether, this has a few unexpected consequences:
1) A world line of constant position, varying coordinate time, will not generally represent a timelike world line (this is a consequence of the impossibility of a rigid ruler per Herglotz-Noether).
2) To produce a valid coordinate chart (e.g. to avoid multiply labeling events), the coverage of the coordinates may be very limited. For a zigzag trajectory, they will be limited to a world tube around the origin.
