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The Representation of Time and Change in Mechanics

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The Representation of Time and Change in Mechanics GORDON BELOT THE REPRESENTATION OF TIME AND CHANGE IN MECHANICS ABSTRACT: This chapter is concerned with the representation of time and change in classical (i.e., non-quantum) physical theories. One of the main goals of the chapter is to attempt to clarify the nature and scope of the so-called problem of time: a knot of technical and interpretative problems that appear to stand in the way of attempts to quantize general relativity, and which have their roots in the general covariance of that theory. The most natural approach to these questions is via a consideration of more clear cases. So much of the chapter is given over to a discussion of the representation of time and change in other, better understood theories, starting with the most straightforward cases and proceeding through a consideration of cases that lead up to the features of general relativity that are responsible for the problem of time. Keywords: Classical Mechanics; General Relativity; Symmetry; Time Draft version 2.0. Forthcoming in J. Butterfield and J. Earman (eds.), Handbook of the Philosophy of Physics. North-Holland. CONTENTS 1 INTRODUCTION . 1 2 HAMILTONIAN AND LAGRANGIAN MECHANICS . 7 2.1 The n-Body Problem . 7 2.2 The Hamiltonian Approach . 8 2.3 The Lagrangian Approach . 11 3 SYMPLECTIC MATTERS . 14 3.1 Preliminaries . 14 3.2 Symplectic Manifolds . 17 3.3 Presymplectic Manifolds . 19 3.4 Symplectic Structures and Quantization . 21 4 LAGRANGIAN FIELD THEORY . 22 4.1 The Lagrangian Approach . 25 4.2 The Structure of the Space of Solutions . 28 4.3 Symmetries and Conserved Quantities . 29 5 TIME AND CHANGE IN WELL-BEHAVED FIELD THEORIES . 30 5.1 The Lagrangian Picture . 31 5.2 The Hamiltonian Picture . 33 5.3 Relation between the Lagrangian and Hamiltonian Pictures . 36 5.4 Time and Change . 37 5.5 Overview . 38 6 COMPLICATIONS . 39 6.1 Singular Dynamics . 40 6.2 Gauge Freedom . 45 6.3 Time-Dependent Systems . 59 7 THE PROBLEM OF TIME IN GENERAL RELATIVITY . 64 7.1 The General Covariance of General Relativity . 64 7.2 The Problem of Time . 77 7.3 Finding Time in General Relativity . 80 ACKNOWLEDGEMENTS . 89 BIBLIOGRAPHY . 89 THE REPRESENTATION OF TIME AND CHANGE IN MECHANICS 1 If time is objective the physicist must have discovered that fact, if there is Be- coming the physicist must know it; but if time is merely subjective and Being is timeless, the physicist must have been able to ignore time in his construction of reality and describe the world without the help of time. If there is a solution to the philosophical problem of time, it is written down in the equations of mathe- matical physics. Perhaps it would be better to say that the solution is to be read between the lines of the physicist’s writings. Physical equations formulate specific laws . but philosophical analysis is concerned with statements about the equations rather than with the content of the equations themselves. —Reichenbach.1 For many years I have been tormented by the certainty that the most extraor- dinary discoveries await us in the sphere of Time. We know less about time than about anything else. —Tarkovsky.2 1 INTRODUCTION This chapter is concerned with the representation of time and change in classical (i.e., non-quantum) physical theories. One of the main goals of the chapter is to attempt to clarify the nature and scope of the so-called problem of time: a knot of technical and interpretative problems that appear to stand in the way of attempts to quantize general relativity, and which have their roots in the general covariance of that theory. The most natural approach to these questions is via consideration of more clear cases. So much of the chapter is given over to a discussion of the representation of time and change in other, better understood theories, starting with the most straightforward cases and proceeding through a consideration of cases that pre- pare one, in one sense or another, for the features of general relativity that are responsible for the problem of time. Let me begin by saying a bit about what sort of thing I have in mind in speak- ing of the representation of time and change in physical theories, grounding the discussion in the most tractable case of all, Newtonian physics. As a perfectly general matter, many questions and claims about the content of a physical theory admit of two construals—as questions about structural features of solutions to the equations of motion of the theory, or as questions about structural features of these equations. For instance, on the one hand time appears as an aspect of the spacetimes in which physics unfolds—that is, as an aspect of the background in which the solutions to the equations of the theory are set. On the other, time is represented via its role in the laws of physics—in particular, in its 1[1991, pp. 16 f.]. 2[1991, p. 53]. 2 GORDON BELOT role in the differential equations encoding these laws. So questions and claims about the nature of time in physical theories will admit of two sorts of reading. Consider, for instance, the claim that time is homogeneous in Newtonian physics (or, as Newton would put it, that time flows equably). There are two sorts of fact that we might look to as grounding this claim. 1. There is a sense in which time is a separable aspect of the spacetime of Newtonian physics and there is a sense in which time, so considered, is homogenous.3 2. The laws of the fundamental-looking theories of classical mechanics (e.g., Newton’s theory of gravity) are time translation invariant—the differential equations of these theories do not change their form when the origin of the temporal coordinate is changed—so the laws of such theories are indifferent to the identity of the instants of time. In the Newtonian setting, these two sorts of considerations mesh nicely and pro- vide mutual support: there is a consilience between the symmetries of the laws and the symmetries of spacetime. But in principle, the two sorts of considera- tion need not lead to the same sort of answer: one might consider a system in Newtonian spacetime that is subject to time-dependent forces; or one could set the Newtonian n-body problem in a spacetime which a featured a preferred instant, but otherwise had the structure of Newtonian spacetime. And as one moves away from the familiar setting of Newtonian physics, it becomes even more important to distinguish the two approaches: in general relativity, the laws have an enormous (indeed, infinite-dimensional) group of symmetries while generic solutions have no symmetries whatsoever. In discussing the representation of time and change, this chapter will focus on structural features of the laws of physical theories rather than on features of par- ticular solutions. To emphasize this point, I will say that I am interested in the structure of this or that theory as a dynamical theory. I will approach my topics via the Lagrangian and Hamiltonian approaches to classical theories, two great over-arching—and intimately related—frameworks in which such topics are naturally addressed.4 Roughly speaking, in each of these approaches the content of the equations of a theory is encoded in certain structures 3(Neo)Newtonian spacetime is partitioned in a natural way by instants of absolute simultaneity, and time can be identified with the structure that the set of these instants inherits from the structure of spacetime: time then has the structure of an affine space modelled on the real numbers—so for any two instants, there is a temporal symmetry which maps one to the other. 4Why pursue our question within the realm of Lagrangian and Hamiltonian mechanics rather than working directly with the differential equations of theories? Because the benefits are large: these over- arching approaches provide powerful mathematical frameworks in which to compare theories. And because the costs are minimal: almost every theory of interest can be put into Lagrangian or Hamil- tonian form, without any obvious change of content. And because it leads us where we want to go: current attempts to understand the content of classical physical theories are necessarily shaped by ef- forts to construct or understand deeper, quantum theories; and it appears that a classical theory must be placed in Lagrangian or Hamiltonian form in order to be quantized. THE REPRESENTATION OF TIME AND CHANGE IN MECHANICS 3 on a space of possibilities associated with the theory.5 In the Lagrangian approach the featured space is the space of solutions to the equations of the theory, which for heuristic purposes we can identify with the space of possible worlds allowed by the theory.6 On the Hamiltonian side, the featured space is the space of initial data for the equations of the theory, which we can in the same spirit identify with the space of possible instantaneous states allowed by the theory. In Newtonian mechanics, the reflection within the Lagrangian framework of the time translation invariance of the laws is that the space of solutions is itself invari- ant under time translations: given a set of particle trajectories in spacetime obeying Newton’s laws of motion, we can construct the set of particle trajectories that re- sult if all events are translated in time by amount t; the latter set is a solution (i.e., is permitted by the laws of motion) if and only if the former set is; furthermore, the map that carries us from a solution to its time translate preserves the structure on the space of solutions that encodes the dynamics of the theory.
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