TiP 2: Classical Mechanics and Clocks

By | 2019-10-13

In the last post, we discussed some ideas of classical mechanics, which is a theory that describes the motion of large objects like stars or billiard balls. We found that time in this theory is a Newtonian time, that is, an external parameter labeling snapshots (configurations, or relative positions) of some objects. Let’s call the collection of objects the physical system or just the system. Also, we said that we don’t really want some “divine” external parameter that we don’t understand, but rather something that is internal to the system. Defining time as such an internal measure of relative motions can be achieved by comparing the motion of one part of the system with another part.

Figure 1: A runner on a track (top view). The yellow thing next to the track represents an analogue clock. The hand (not the arms or legs!) of the clock labels the position of the runner on the track — it can be used as a (time) parameter for the motion of the runner.

This is also what we actually do when we measure time: We compare the dynamics of our system of interest (e.g. the motion of the runner on the track) to the motion of another part of the universe, which we may call the “clock” — it is the physical thing that actually provides a measure for the abstract notion of time. Ignoring a few subtleties for now, we can say that a clock is an object whose state changes continuously as the system that we are looking at changes. For example, the hand of an analogue clock changes continuously its position while a runner runs her run, hence the position of the hand can be used to parametrize the run: Each position of the hand corresponds to a position of the runner along the track, and the continuous change of the hand corresponds to a continuous change of the position of the runner. This is shown schematically in figure 1.

For a something to be a clock that provides a time parameter, we need to know where its “hand” is. In classical mechanics1 this is not a problem, as in principal the position of each particle/object/something and its velocity is exactly known. Also, a good clock is little influenced in its motion by the environment. A normal stopwatch can be a good clock for a run, but a pendulum clock that the runner has to carry during a run may be a pretty bad clock because it gets shaken along the way — both clocks will show a time, but the time from the stopwatch is likely in much better agreement to the time that the clocks show which are used for the definition of a second. However, in principle both the time from the stopwatch and the time from the well-shaken pendulum clock may be used to parametrize the position of the runner. We only choose to use good clocks because good clocks show the same time everywhere (as long as the theory of relativity is irrelevant) and their times can thus directly be compared. As all good clocks show a similar time reading, we are lead to believe that there is some fundamental quantity that we are measuring. Time certainly is very relevant for us, as we compare everything that we experience (at least) to the motion of the earth around its axis and around the sun, and we use this daily or yearly rhythm to organize our life. Nevertheless, time actually is a rather arbitrary parameter! It only looks like a fundamental quantity because we are able to construct good clocks that all show a very similar time parameter.

The view that time needs to be defined via a clock and that, instead of assuming an external time parameter, the clock should be treated explicitly as part of the problem, has become relatively popular in recent years. This view is connected with the idea that there cannot be a Newtonian time for the whole universe (or for a closed system), but time can only be defined internally via a comparison of a dynamics of some part of the universe with another part, which serves as a clock.2

We can illustrate the idea of time being defined via an internal clock with balls on a billiard table, as shown in figure 2. Some snapshots of the motion of these balls are shown: We have a white ball that gets hit by the cue, which hits a blue ball, which then hits the wall of the table. However, there is also an innocent looking orange ball which moves along the table.3 The direction along which it is moving is, say, the \(y\)-direction, and the coordinates of the orange ball are two numbers \((x_{\rm o},y_{\rm o})\). While \(x_{\rm o}\) does not change, \(y_{\rm o}\) will change during the motion of the white or blue balls. Let us also assume that the orange ball moves with constant speed \(v_{\rm o}\).4 We can use the orange ball as our clock for the motion of the other balls by defining a time parameter \(t := y_{\rm o} / v_{\rm o}\). This parameter has units of time, but because we assume that \(v_{\rm o}\) does not change we could also use \(y_{\rm o}\) directly as parameter — although this parameter has units of a distance, it can be used exactly like the time parameter \(t\) to label the motion of the other balls. For example, instead of describing the motion of the white ball with the two numbers \((x_{\rm w},y_{\rm w})\) as function of \(t=y_{\rm o} / v_{\rm o}\), we can write them directly as function of \(y_{\rm o}\), that is, \((x_{\rm w}(y_{\rm o}),y_{\rm w}(y_{\rm o}))\). In this way, we have eliminated the mysterious external time parameter \(t\) and replaced it with an internal parameter of the system, that is, either with \(y_{\rm o} / v_{\rm o}\) or just with \(y_{\rm o}\).5

Using an internal picture is important for understanding the meaning of time. Time is not something additional, but a parameter which describes the correlation of motions: It describes how different parts of a system (or of the universe) move relative to each other, and it can be eliminated by explicitly referring the motion of one part of the system to the position and velocities of the other part (used as the clock). Moreover, if we look at the more fundamental theory of the motion of particles, called quantum mechanics, we will see that such a single parameter may not even exist! Instead, it is an approximation that works whenever the clock that we choose effectively obeys the rules of classical mechanics. Hence, it is central to understand that time is a kind of correlation of motions, because in quantum mechanics time in the usual sense does not exists, but only these correlations.

To summarize, we found that

  • an external time parameter can replaced with an internal part of the system of interest,
  • this internal part is a “clock” in a very general sense,
  • a time parameter describes the correlations of motion of different parts of the system, and
  • these correlations are more fundamental than the time parameter.

Some semi-philosophical remarks:

After this discussion you might feel a bit uneasy and ask the question: If time can only defined by comparison of different parts of a physical system, then the universe as a whole should not have a time parameter, shouldn’t it? And if it doesn’t, then why is there a dynamics or motion at all?

It seems indeed to be the case that a theory of the whole universe cannot have a time parameter (more about this will come in future posts). However, the question where dynamics comes from if there is no time parameter may be misleading, because it is based on an external view of the world. Such an external view works for balls on a billiard table, where we could have just watched the motion of the balls and used the hand of a wall clock as parameter — we need not consider the clock to be part of the system. But imagining the whole universe and an additional external clock “next to it” to parametrize changes in the universe is not so clever.

But we do experience something like a time parameter, right? A way to think about what we experience is to say that there is motion (in a very general sense) and because we are a part of the universe, and because we look the world on a scale in which classical mechanics is applicable, we perceive time the way we do. We are little subsystems of the universe which feel a passage of time by comparing the motion/changes of all kinds of things that we perceive. The important point is that we view the universe from within, hence a good concept of time has to account for the fact that our view is a relative, or relational view of the world.

Finally, the idea that time is something external that just “flows” is essentially equivalent to stating that it cannot be understood. Thus, eliminating the concept of an external time and replacing it with the ideal of internal correlations of parts of a physical system is certainly very appealing for a researcher like me who believes that nature is accessible to physical reasoning.

  1. In the more fundamental theory of the world known as quantum mechanics, this is in fact a problem.
  2. A name that you might have heard in this respect is Julian Barbour, a British scientist who has done some interesting research but who also spends some effort in making these ideas accessible to the general public. If you want to learn more about time in classical mechanics, his works are a good reference, see e.g. the article The Nature of Time
  3. If you wonder where the ball is coming from, then, well, don’t. It’s just an illustration. If you don’t like using the orange ball as a clock, then you can also use the motion of your favorite planet around the sun, it works the same way.
  4. This means that the ball moves without friction or that there is some mechanism (e.g. inside the ball) which keeps the velocity constant.
  5. Even if the orange ball hits the wall and changes its direction of motion, we can still keep it as clock by counting the number of wall hits (like the minute hand of a clock counts the number of cycles of the second hand).

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