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Among the (several) descriptions of an "inertial frame" (in the context of the Special Theory of Relativity),
those by John Norton and by Wolfgang Rindler appear especially concise:

and

Are these two descriptions equivalent ?
(even considering their brevity ...)

Specificly:
Given a congruence $\mathfrak C$ of timelike geodesics in a flat region of spacetime,
i.e. (a spacetime-filling) family $\mathfrak C \equiv \{ \mathcal P_n \}$ of world lines which are

  • each timelike,
  • each geodesic (i.e. each straight in terms of Lorentzian distance, resp. Synge's world function, resp. spacetime intervals),
  • non-intersecting (such that there is no event belonging to two or more world lines) and
  • space(time)-filling (such that each event of the given region belongs to exactly one world line)

are any two of those world lines necessarily "sitting still" (or "running in parallel") to each other, in the sense that

$\forall \, \mathcal P_a, \mathcal P_b \in \mathfrak C, \forall \varepsilon_j, \varepsilon_k \in \mathcal P_a \, : $
$\exists \, \varepsilon_u, \varepsilon_v, \varepsilon_x, \varepsilon_y \in \mathcal P_b :$
$s^2[ \, \varepsilon_j, \, \varepsilon_u \, ] = s^2[ \, \varepsilon_j, \, \varepsilon_v \, ] = s^2[ \, \varepsilon_k, \, \varepsilon_x \, ] = s^2[ \, \varepsilon_k, \, \varepsilon_y \, ] = 0$ and
$s^2[ \, \varepsilon_u, \, \varepsilon_v \, ] = s^2[ \, \varepsilon_x, \, \varepsilon_y \, ] \neq 0$

?

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  • $\begingroup$ In my opinion, the Rindler quote is incomplete since there's no explicit reference to geodesic motion---being at relative rest is not enough. $\endgroup$
    – robphy
    May 27 '20 at 16:55
  • $\begingroup$ @robphy: "the Rindler quote [...] being at relative rest is not enough" -- Clearly Rindler's phrase "sitting still in space relative to each other" requires/presumes further definition. But outright, it appears distinctly named from just "being pairwise non-intersecting (a.k.a. non-coincident)", and different from just "being pairwise rigid", e.g. in the sense of chronogeometric rigidity (Synge, GR, p. 108). $\endgroup$
    – user12262
    May 27 '20 at 17:19
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I would say that not only are the definitions not equivalent, they are not even correct. A frame of reference means the reference matter which is used to define a coordinate system. For example we talk of the Earth frame, or if we are in a car, we will use the car as a reference frame. Norton's definition does not even refer to reference matter, but only to the mathematical construct of a geodesic, and Rindler does not refer to actual matter, but to an idealisation which only exists in thought.

Notwithstanding whether the definitions are correct, the fact that one is purely mathematical and the other invokes a notion of physical matter means that they are not equivalent. Mathematics is distinct from physics. A frame of reference is the means by which one relates the mathematical structure of relativity to physical observation, and depends for its definition on the physical structure which one uses as the basis for measurement.

Another issue is that the concept of a geodesic is only needed in general relativity. There is no point in using it in special relativity in which the geodesics are simply straight lines.

It is perfectly possible to give a succinct definition of an inertial frame without using mathematics to create the kind of obfuscation seen in these "definitions". This requires only the application of Newton's first law. For precision one should first define

  • An inertial object is one such that, in any reference frame, the effect on its motion due to contact interactions with other matter is negligible.

Then one defines

  • An inertial reference frame is one in which inertial bodies remain at rest or in uniform motion.

These definitions apply in the general theory of relativity, not just the special theory. They apply locally (an inertial frame is a local concept), and one should recognise that special relativity only applies to physics as local approximation.

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  • $\begingroup$ "A frame of reference means the reference matter [...]" -- Allright (preliminary score: +1), though I'd say: it means a set of material, identifiable participants and their mutual geometric (incl. kinematic) relations or some equivalent. "Norton's definition does not even refer to reference matter [...]" -- I'd accept "a timelike curve" as suitably equivalent to "a point particle", etc. "Newton's first law [...] effect on motion due to contact interactions" -- I've focused on Norton's and Rindler's definitions for being purely geometric/kinematic. [contd.] $\endgroup$
    – user12262
    May 27 '20 at 20:41
  • $\begingroup$ "Then [...]: An inertial reference frame is one in which inertial bodies remain at rest or in uniform motion." -- Apparently, that's not "the reference matter" as such, but something, "in which" it's characterized. (Preliminary score: 0.) Anyways: Your present answer version is utterly missing what I was hoping to learn from asking my question ... I might rephrase: Given two distinct timelike geodesic congruences in the same spacetime region, are they guaranteed disjoint? (Given two distinct sets of "particles sitting still ...", they are disjoint.) $\endgroup$
    – user12262
    May 27 '20 at 20:42
  • $\begingroup$ Yes, the particles in Rindler's definition follow a congruence in Norton's. This does not alter the facts that particles are not a congruence, that neither is a concise definition, because both require elaboration before they make sense, and that both miss the point of what a reference frame actually is. $\endgroup$ May 27 '20 at 21:14
  • $\begingroup$ btw, a reference frame does not have kinematic relations, since kinematic relations are defined relative to it. One might say the same about geometric relations, since geometry depends on measurement, and measurement is relative to the reference frame. One can define minimal conditions for a reference frame, e.g. an origin, a means of measuring time, a ruler, and three spacial axes, but that is, in itself, somewhat less than geometry. When I say reference matter, I do indeed mean "material, identifiable participants". $\endgroup$ May 27 '20 at 21:23
  • $\begingroup$ "Yes, the particles in Rindler's definition follow a congruence in Norton's. [...] particles are not a congruence" -- Fair enough: equivalence is not identity. "both [Norton's and Rindler's descriptions] require elaboration" -- Sure. Which word or which phrase doesn't?? (Among the few exceptions: the words "same" and "distinct".) "both miss the point [...]" -- I beg to differ. (And perhaps at least Norton, still, too.) [contd.] $\endgroup$
    – user12262
    May 27 '20 at 21:45

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