Are John Norton's and Wolfgang Rindler's notions of "inertial frame (in special relativity)" equivalent?

Question

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:

• "In special relativity, an inertial frame of reference is a congruence of timelike geodesics.",
  John D. Norton, "General covariance and the foundations of general relativity: eight decades of dispute" (1993), p. 837

and

• "An inertial frame is simply an infinite set of point particles sitting still in space relative to each other.",
  Wolfgang Rindler (2011), Scholarpedia, 6(2):8520, "Special relativity: kinematics".

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
$s^2[ \, \varepsilon_u, \, \varepsilon_v \, ] = s^2[ \, \varepsilon_x, \, \varepsilon_y \, ] \neq 0$

?

Answer 1

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.