How to identify a "measuring rod", and how to compare separated "measuring rods" with each other?

Question

The notion of "measuring rod" has appeared in PSE here and there, and outside PSE as well.

As far as I understand (and as perhaps all who refer to this notion do agree on), important constituents of any one "rod" are "its two ends", being two distinct and (in general) separate material points (or principal identifiable points, or for short: participants).

Therefore the following questions remain to identify measuring rods among all rods, and to explain their use:

1. Do all rods, i.e. all pairs of distinct participants, in all trials, make up a measuring rod, constituting "its ends"?,

Or else:

2. How should be determined for a given rod, i.e. for a given particular pair of distinct participants, in a particular trial under consideration, whether these two had made up a measuring rod, constituting "its ends", or not?

And:

3. How should be determined whether two distinct measuring rods, in one particular trial, had been equal to each other, or not; i.e. especially if they had only at most one end in common, or if only at most one end (of one of the two measuring rods being considered) had been coincident with one end of the other?

Answer 1

Within the theory of relativity there exist several different notions of how to characterize a "measuring rod" by a relation between its two "ends"; and correspondingly there are different notions of two distinct participants having been "rigid" to each other.

In listing some of those notions it is of course possible at least to discuss the special case that there are no requirements on two "ends" at all, aside from having been distinguishable. In this sense question 1 can be answered with: "It depends."; but typically and more properly there are additional specific requirements which have to be satisfied by two distinct participants to be called the two "ends of one measuring rod", and which would not be satisfied by just any two distinct participants whatsoever.

The applicable requirements on any two participants under consideration can broadly be classified as either "proper" (i.e. in terms of observational data collected by these two participants themselves) or else "improper" (i.e. only in terms of observational data which others collected about them).

There is obviously only very specific observational data (relevant within the RT) which two participants are capable of collecting themselves about each other, namely foremost (if not even exclusively): for any one separate signal indication stated by one participant whether there had been a corresponding reception indication by the other, or not. On basis of this observational data, given in any sufficiently extended trial, two participants under consideration can be characterized as "ends" of a "proper measuring rod" of the following (ever more inclusive) types:

• commutative measuring rod: both ends (in the following $A$ and $B$) observe all of each other's indications (each as signal indication and/or as reception indication), for each signal indication within the trial the completion of the signal round trip (a.k.a. "ping") is observed, and the durations of these signal roundtrips ("ping durations") of either end are constant non-zero and both (constants) are equal to each other:
  $\tau A_{\circ}^{\circledR B} = \tau B_{\circ}^{\circledR A} := \text{constant} \neq 0$.
  Two distinct commutative measuring rods (one constituted of ends $A$ and $B$, the other constituted of ends $J$ and $K$, at least one of whom is distinct from both $A$ and $B$) are equal to each other if and only if the ping durations of both pairs are equal:
  $\tau A_{\circ}^{\circledR B} = \tau B_{\circ}^{\circledR A} = \tau J_{\circ}^{\circledR K} = \tau K_{\circ}^{\circledR J}$.

• bilateral measuring rod: both ends observe all of each other's indications, for each signal indication within the trial the completion of the signal round trip is observed, and the ping durations of either end are (separately) constant non-zero (but not necessarily equal):
  $\tau A_{\circ}^{\circledR B} := \text{constant} \neq 0, \quad \tau B_{\circ}^{\circledR A} := \text{constant} \neq 0$.
  Two distinct bilateral measuring rods (pair $A$ and $B$, and pair $J$ and $K$, at least one of whom is distinct from both $A$ and $B$) are equal to each other if and only if their corresponding ping durations can be equally matched:
  $\tau A_{\circ}^{\circledR B} = \tau J_{\circ}^{\circledR K}, \quad \tau B_{\circ}^{\circledR A} = \tau K_{\circ}^{\circledR J}$.

• unilateral measuring rod: both ends observe all of each other's indications, for each signal indication within the trial the completion of the signal round trip is observed, and the ping durations of one end are constant (but not necessarily the ping durations of the other end) and the ping durations of either end are non-zero (i.e. the two ends were never coincident):
  $\tau A_{\circ}^{\circledR B} := \text{constant} \neq 0$, therefore also: $\tau B_{\circ}^{\circledR A} \neq 0$.
  Two distinct unilateral measuring rods (pair $A$ and $B$, and pair $J$ and $K$, at least one of whom is distinct from both $A$ and $B$) are equal to each other if and only if their (unilateral) constant ping durations are equal:
  $\tau A_{\circ}^{\circledR B} = \tau J_{\circ}^{\circledR K} := \text{constant} \neq 0$.

Commutative, bilateral, or unilateral measuring rods are also called "chronometrically rigid" to each other, in a commutative (or equivalent), bilateral (or mutual), or unilateral sense, respectively. A further strengthened notion is the

• measuring rod at rest: as a commutative measuring rod (ends $A$ and $B$) together with three distinct additional participants ($J$, $K$, $Q$) such that all ten pairs among these five distinct participants constitute ends of a commutative measuring rod, and these five participants are flat to each other (i.e. the Cayley-Menger determinant in terms of the ping durations of these ten pairs of ends vanishes).
  (While here there are three additional participants required in order to define this notion for a given pair of participants $A$ and $B$, the observational data collected by $A$ and $B$ about each other, as well as about the additional participants, is nevertheless properly employed. Therefore any "measuring rod at rest" can be classified among the proper measuring rods.)

Further relations between pairs of distinct participants, in terms of observational data collected by these two participants themselves, are:

• "at least some sort of actual rod": both ends observe all of each other's indications, and for each signal indication within the trial the completion of the signal round trip is observed. The only condition on the corresponding ping durations of either end is, that they don't vanish; i.e. the two ends were never coincident: $\tau A_{\circ}^{\circledR B} \neq 0$, therefore also: $\tau B_{\circ}^{\circledR A} \neq 0$.
  Two pairs of ends (pair $A$ and $B$, and pair $J$ and $K$) which constitute two distinct "sort of actual" rods can be considered equal to each other insofar as they do satisfy at least this condition equally, rather than not.

In the remaining weakest cases of "measuring rod" in a proper sense may be classified as

• any possibly degenerate rod: both ends observe all of each other's indications, and for each signal indication within the trial the completion of the signal round trip is observed. There is no condition on the corresponding ping durations (therefore they may vanish; i.e. the two ends may have been coincident).
  (Any possibly degenerate rods may be compared to each other as described above.)

• hardly even two ends to speak of: at least one end does not observe the completion of signal round trips with respect to the other, within the trial under consideration.

Additionally various improper notions of "measuring rod" may be defined and considered, i.e. in terms only of observations which other participants collected about the pair under consideration. For instance:

• Born-rigid measuring rod: the two ends, $A$ and $B$, are (without interruption) met (generally in passing) by the ends of a family of "measuring rods at rest", such that those "measuring rods at rest" are equal to each other (i.e. in the sense of also constituting a family of equal "commutative measuring rods", as defined above) for which the indication of one end having been met and passed by $A$ had been simultaneous to the indication of other end having been met and passed by $B$.
  Two distinct Born-rigid measuring rods are equal if their referenced "measuring rods at rest" are equal between the two corresponding families.
  (Any "measuring rod at rest" is also a "Born rigid measuring rod" by definition. But any "commutative measuring rod" is not necessarily a "Born rigid measuring rod"; and any "Born rigid measuring rod" is not even necessarily "hardly even two ends to speak of".)

• Or, as a case added for illustration: the two ends, $A$ and $B$, are (without interruption) met (generally in passing) by the ends of a family of "commutative measuring rods", such that all distinct such ends met and passed by $A$ and all distinct such ends met and passed by $B$ constitute "commutative measuring rods". (For given pairs of participants, $A$ and $B$, there may exist several distinct suitable families of "commutative measuring rods".)
  Call the pair $A$ and $B$ and pair $J$ and $K$ equal to each other (in some particular trial, due to having satisfied the described relation) if the maximal ("longest") "commutative measuring rod" among all those familiy members having met $A$ or $B$ is equal to the maximal ("longest") "commutative measuring rod" among all those familiy members having met $J$ or $K$.

Answer 2

A measuring rod is an object that has a constant proper length.

Start with the simple example of a ruler in Euclidean space. If we place one end of the ruler at the origin then the other end will be at some point $(\Delta x, \Delta y, \Delta z)$, where obviously the position will depend on the length and orientation of the ruler. The length of the rod, $s$, is given by Pythagoras' theorem:

$$ s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 \tag{1} $$

Now switch to special relativity. The ends of the rod are now at points in spacetime, $(t, x, y, z)$. Once again we place one end of the rod at the origin so the other end of the rod is at $(\Delta t, \Delta x, \Delta y, \Delta z)$. The length of the rod is now given by the Minkowski metric:

$$ s^2 = -\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 \tag{2} $$

This quantity $s$ is called the proper length and it's an invarient, that is all observers will agree on its value.

In the rest frame of the rod both ends have the same time coordinate so $\Delta t = 0$. In that case equations (1) and (2) for $s$ are the same i.e. in the rest frame of the measuring rod its proper length is the same as its length in Euclidean space, hence the name.

In general relativity life is more complicated because the proper length has to be calculated by integrating the expression:

$$ ds^2 = g_{\mu\nu}dx^\mu dx^\nu $$

where $g_{\mu\nu}$ is the metric that describes the geometry of whatever spacetime you're working in.

Now back to the specific points you raise in your question:

In your questions (1) and (2) you mention pairs of distinct participants. If we take this to mean pairs of spacetime points then those pairs form a useful measuring rod if they have a constant proper length. Basically this means there has to be an inertial frame in which both points are stationary and remain stationary. If the points really are the ends of a rod then of course this condition is satisfied in the rest frame of the rod (assuming the rod isn't made of something stretchy).

Your question (3) is somewhat subtle. If we can measure the positions of the ends of the two rods in our coordinate system then we can calculate the proper length, and if we calculate the same proper length for both rods we would consider them to be the same length. But if the rods are moving with respect to each other then they have different rest frames and in all but one inertial frames their "length" calculated with equation (1) will be different. The frame where they have the same length is the frame in which the rods have equal but opposite velocities, but even in this frame the length calculated using (1) won't be the same as the length in the rest frame of either rod.