PAllen said:
Correct.
Rods are typically assumed to be 'born rigid objects'. In this, case, for a world line defining a time axis for an accelerated frame, and assumed not to be rotating, it turns out the Fermi-Normal coordinates match measurements made with born rigid rulers (with one end following the axial world line). If rotation is allowed, there is no possible definition of rigid rods per Herglotz-Noether.
If one introduces the "relative space" of Rizzi and Ruggerio,
https://arxiv.org/abs/gr-qc/0207104, one can gain some insight into the physical experience of space on a rotating disk. The approach differs in who it handles time - instead of trying to define a rigid motion in space-time, one focuses on a different way to separate "space" from "time" in space-time.
This approach is also called by some authors a "quotient manifold", it's the sort of space one gets by eliminating the time dimension in a specific way. One does this by mapping entire lines in space-time to a single point in the abstract space. This is done routinely to separate space from time - any map from space-time to space must map a 4d manifold to a 3d manifold, so it must map lines in the 4d space-time to points in the 3d space. In the R&R approach, rather than focus on creating surfaces of simultaneity which is the usual approach (and one that causes problems on the rotating disk), one instead keeps things simple by mapping the worldlines of particles "at rest" on a rotating disk to single points in the abstract space, the "relative space" of R&R.
The result of this process is a static spatial geometry (assuming the space-time geometry is that of a disk rotating with constant angular velocity, i.e. a stationary rotating space-time), , with a well defined notion of distances between points given by a static spatial metric that corresponds physically to the SI definition of the meter.
SI meter:
The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.
There aren't any special difficulties that I'm aware of in using an older defintion of the meter rod rather than the modern SI definitoin, except for accuracy issues. Both techniques are suitable for measuring the distance between sufficiently nearby points, though the light-defined meters will both be more accurate, and less easily deformed by inertial pseudo-forces than physical meter bars. Both, though, should reach the same limiting concept of idstance over a sufficiently short interval.
We study the space geometry of a rotating disk both from a theoretical and operational approach, in particular we give a precise definition of the space of the disk, which is not clearly defined in the literature. To this end we define an extended 3-space, which we call relative space: it is recognized as the only space having an actual physical meaning from an operational point of view, and it is identified as the 'physical space of the rotating platform'. Then, the geometry of the space of the disk turns out to be non Euclidean, according to the early Einstein's intuition
I make out the "operational" point of view of R&R to simply say that they are using the current SI defintion of the meter, as I mentioned.
I think perhaps the biggest difficulty with this is the notion of accepting that it is sufficient to define the distance between all nearby points in a "space" via some sort of metric. People want to leap ahead into defining distance over longer intervals.
There are several logical ways to proceed, the problem is that authors may use different approaches, which are different enough to give different answers, so there isn't any short way to talk about large-enough distances. The underlying difficulty in most cases is how to handle the problem of simultaneity, along with the approach of how to convert space-time into space+time.
R&R eliminates time right away, the manner in which they do so leads to a certain definition of distance. Fermi-normal coordinates take a different approach, in which one defines surfaces of simultaneity by space-like geodesic radiating from some specific observer. They work well in the vicinity of the observer, but may not work so well globally - they may not respect global symmetries. Cosmologists often base their notion of distance on taking surfaces of constant time - which is probably the most popular approach for non-rotating coordinate systems. But this approach has difficulties in dealing with rotating frames of reference.