The reparametrization of time

Introduction

We saw in the previous chapter that Newton’s Absolute Time can be used to parametrize successive classical mechanical (CM) states of systems under observation (SUOs). We also commented on the fact that Newton was aware that Absolute Time is an idealization, so he defined measured time as ‘relative, apparent, and common time’, which is a ‘. . . sensible and external (whether accurate or unequable) measure of duration’ [Newton, 1687]. In this chapter we look more carefully at this issue. Specifically, following on from our discussion in Chapter 12 on the single dimensionality of the time parameter, we discuss in this chapter how variable this parameter can be in mechanics, that is, investigate how precisely we need to pin time down.

This discussion is of importance to us because it leads to perhaps the deepest question of them all: what is the physical status of time? Is time real or not?

Surprisingly, an important perspective on this question is to be found in the branch of mechanics known as constraint mechanics. Constraints arise naturally in CM whenever there is a mathematical redundancy in the description of some physical system. This happens surprisingly frequently. Suppose a theorist knows that a given SUO has a certain number n of physically observable degrees of freedom, but for some reason that theorist finds it necessary to employ a greater number m of mathematical degrees of freedom (coordinates) to model that system. This happens in all modern field theories. In such a case, constraint mechanics generates just the right number m - n of constraints that effectively eliminate the redundant, unphysical degrees of freedom, resulting in a theory that has the right number n of physical degrees of freedom.

An important example of such a situation occurs in the special relativistic (SR) formulation of the classical point particle. The logic goes as follows. If we started discussing a point particle in non-relativistic (Newtonian) CM, we would use three spatial degrees of freedom, such as the Cartesian position coordinates x,y, and я in an inertial frame F. We would use laboratory time t to label the particle’s instantaneous position in space, relative to our frame of reference F. This is a space-time perspective (note the hyphen), where time t is a parameter.

Images of Time. First Edition. George Jaroszkiewicz.

© George Jaroszkiewicz 2016. Published in 2016 by Oxford University Press.

We shall see in later chapters that Minkowski’s approach to SR [Minkowski, 1908] led to the idea that space and time should be merged into a single entity called Minkowski spacetime (no hyphen), a four-dimensional continuum with a pseudo-Euclidean distance structure or metric. According to Minkowski’s approach, the particle discussed above should be described by a worldline in this four-dimensional continuum. A worldline is a set of events or points in spacetime with four spacetime coordinates {t, x, y, z}, parametrized by some new parameter т that plays the role of an internal time. From this perspective, the SUO appears now to have four degrees of freedom, not three, because t is now regarded not as a parameter but as a dynamical degree of freedom.

This raises the obvious question: how can a Newtonian space-time description that involves three mechanical degrees of freedom x, y, z describe the same physics as an SR description that involves four mechanical degrees of freedom t, x, y, z?

The answer is provided by constraint mechanics: it shows how to accommodate both the space-time perspective and the spacetime perspective. Constraint mechanics is relevant to us in this chapter because we shall need it when we attempt to reparametrize time. We do not need to be discussing SR: the method works equally well for purely Newtonian, non-relativistic systems.

We will follow the analysis given in an influential book on constraint mechanics by Dirac [Dirac, 1964]. It was Dirac’s intention in that book to lay down an approach to quantum gravity that would permit the so-called canonical quantization of Einstein’s GR. Although Dirac did not succeed in giving a consistent quantization scheme for GR, his approach to constraints has been applied successfully to many other situations, including the one we are concerned with now.

 
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