Do bad clocks measure proper time?

Question

It has been claimed several times e.g. here, and here, also here, or (outside PSE) there that

Clocks measure proper time.

Or equivalenty:

proper time is just the time that would be measured by a clock traveling along that timelike curve as its world-line

Or in short:

clocks measure arc-length.

All these referenced claims are apparently without any further qualification of the clocks being considered. In particular, there is no recognizable mentioning of clocks having been "good" (as opposed to having been "bad") in the sense of MTW §1.5. Therefore I like to know:

Do all clocks measure proper time, including bad clocks?

(When answering, also consider applicable follow-up questions such as how "good clocks" and "bad clocks" ought to be distinguished experimentally, at least in principle; or which definition of a "clock" would include "good clocks" as well as "bad clocks".)

Answer 1

My point of view in physics is that, given any concept (in this case, proper time), there are always two notions: (1) the theoretical concept defined in the sense of mathematics, and (2) the experimental concept defined in the sense of experiment. We then hypothesize that these two concepts are equal, and of course, if experiment shows that this is wrong (e.g., by showing that the measured quantity does not obey the expected properties of the theoretical quantity), then our assumption that these two notions were in fact one of the same is wrong, and hence we must go back to the drawing board.

Given this perspective, the mathematical definition of proper time is quite simple. Given a time-like curve in space-time (thought of as the world-line of an observer), the proper time of this curve is the arc-length of this curve (as determined by the space-time metric).

On the other hand, experimentally, proper time is defined by 'how long' (whatever that means) it takes for a certain number of events to occur (see, e.g., the definition of the second), and a clock is a device we use to measure this.

I would then define a good clock as a clock whose measurements agree with the theoretical concept of proper time. Any other clock is a bad clock.

Given the definition posed of a bad clock in the comments

it makes the world lines of free particles through the local region of spacetime look curved

I would like to show that this is equivalent to the statement that a clock is not good (so that the two definitions, the one I gave and the one here, are in fact equivalent). Here, I am going to interpret the phase "makes . . . spacetime look curved" as meaning that the measurements given by the clock that moves along a geodesic would imply that it is not actually a geodesic.

Locally (the word "local" in the above definition is key), a time-like curve between two points is a geodesic iff it maximizes the arc-length of all time-like curves between those two points. If this maximum arc-length is $\tau _0$, then from this it follows that the clock would show that the curve is not a geodesic iff the time it measures is less than $\tau _0$. However, $\tau _0$ is precisely the proper time between the two points, which means that the clock would show that the curve is not a geodesic iff it measures a time less than $\tau _0$. This finishes the proof of equivalence.

Answer 2

In contrast to coordinate time, proper time is independent from any geodesics geometry. Any clock is OK as long as it is in the same frame as the object whose proper time is measured.

In order to recover the proper time information, the observer must ensure synchronization of his own clock with the clock of the observed object. Thus, your problem might be not a problem of bad clock but of insufficient synchronization.