Trying to reformulate the question I had asked Lubos became a bit long-winded, so I made it a separate post
Let’s consider the much simpler problem of a photon travelling in the vacuum. It is legitimate to write an equation like to predict where the photon is, meaning that the distance measured from the starting point is proportional both to elapsed time and speed. We can write a very similar equation describing the rotation of earth, something like . Finally, we can also consider a car travelling at constant speed, where the equation would be something like .
Are the time t or position x written in these equations the same? That is really the meaning of the question I asked. The fifth-grader answer would be something like “yes, we just use a clock and a ruler in both cases”. Apparently, this is also Lubos’ answer. And, indeed, for short enough durations, this works quite well. So, conversely, we may be tempted to define time or duration using one of these phenomena. For instance, we currently define distance using the first equation (check the definition of the metre). And we historically defined time based on cosmic events like earth rotation.
Scales tends to be fatal to simple linear laws
But obviously, for large enough values of t, a number of things will happen. For example, the earth rotation is now known to not be exactly regular compared to, say, cesium clocks. So “earth time” will not remain very well aligned with the kind of tool we now use to measure time. Furthermore, because the earth surface is not flat, the x we used for the car will very soon turn out to be quite different from the x for the photon, as is obvious if you consider where the photon and the car would be after 10000km (cars, unlike photons, don’t fly after all).
It is very tempting to say that the earth rotation is irregular and that’s it. But a time defined based on this particular physical process is still the only one that allows us to keep the simplest form for the second relation. For anything related to earth, it’s basically “a better time” than a definition based on cesium, in the sense that it keeps laws of physics simpler… This remark is just another formulation of one of the key insights of general relativity, that you can really pick the time or space coordinate you want, that there is no preferred one, and that you can still write physics laws with that somewhat arbitrary system of coordinates.
In my examples, some of the coordinates I used are not obviously definitions of time until you relate them to time with some law. Cosmic periodic events, such as earth rotation, may be chosen as a definition of time (day, month, year, and so on). I can relate this definition to another, for example counts of individual “ticks” in a cesium clocks. At small scale, the relation will be approximately linear. However, at larger scale, the law connecting this and that definition of time becomes essentially arbitrary, so I need some arbitrary calibration, something like , to relate any two definitions.
How do you recognize that a physical measurement measures time?
Now I can ask the question backwards. How do I recognize that a particular physical measurement can be used as a definition of time? Or mass? Or position? To better understand what the question means, imagine that you are given an experiment, with measurement results (e.g. graphs, tabular results, …), and you need to determine which one is time, which one is mass, which one is speed, which one is position, and so on. How would you do that?
If there is any difficulty defining time or space coordinates for something as simple as a constant speed equation, can we blissfully ignore the problem for the kind of equations you find in Lubos’ papers? Can we say: “I just pick up a scale and put my blackhole on top of it” as Lubos asked me to do?