Part 3 — CHAPTER XXXV THE IMPORTANCE OF SPACE-TIME, AND THE PRINCIPLE OF ACTION

IN the course of this book we have seen how a number of difficulties were overcome as soon as we recognised the existence of four-dimensional space-time in lieu of the world of separate space and time of classical science. There are still other aspects of the problem, however, and we shall now mention them briefly.

In the first place, all vectors such as velocities and forces, and all tensors such as Maxwell’s stresses of the electromagnetic field, erstwhile expressed in three-dimensional space, must now be extended and supplemented with additional components so as to yield vectors and tensors in four-dimensional space-time. These new components do not define new physical magnitudes unknown to classical science; in the general case they give us well-known magnitudes, such as work, power, and so on. As a case in point, let us consider electric and magnetic intensities. In classical science these were entirely distinct magnitudes. But when we represent these intensities in terms of space-time, we find that they are given by the various components of one same space-time tensor ; the time components give the electric force, while the space components yield the magnetic force. The advantage of the new point of view is chiefly to show us that what we once considered different entities now reveal themselves as the different components of the same space-time existent. The reason for the observed connections between these different entities then becomes apparent.

In particular, let us examine the case of the equations of electrodynamics. In the study of electricity and magnetism we may consider phenomena in which conditions do not vary as time passes by; the electric charges and the magnets remain at rest, and currents flowing in fixed wires do not vary in intensity. Conditions are then termed stationary; it is as though time played no part. The laws which govern this type of phenomena were discovered empirically over a century ago, and were expressed mathematically in terms of spatial vectors. The problem of ascertaining how electric and magnetic phenomena would behave when conditions ceased to be stationary was one that could not be predicted; further experimental research work was necessary before the general laws could be obtained. Even so, the difficulties were considerable, and it needed Maxwell’s genius to establish the laws from the incomplete array of experimental evidence then at hand. All this work extended over nearly a century; it was slow and laborious.

Yet, had men realised that our world was one of four-dimensional Minkowskian space-time, and not one of separate space and time, things would have been very different. By extending the well-known stationary laws to four-dimensional space-time, through the mere addition of time components to the various trios of space ones, we should have written out inadvertently the laws governing varying fields, or, in other words, we should have constructed Maxwell’s celebrated equations. Electromagnetic induction, discovered experimentally by Faraday, the additional electrical term introduced tentatively by Maxwell, radio waves, everything in the electromagnetics of the field, could have been foreseen at one stroke of the pen. A century of painstaking effort would have been saved.[121]

It is also interesting to note that the four separate relations which constitute Maxwell’s equations of electromagnetics are now merged into two; and the cumbersome aspect of the equations gives place to forms of great simplicity.

Still another simplification which the discovery of space-time has conferred upon us is to be seen when we study the principles of the conservation of energy and of momentum. We find that these two distinct principles constitute in reality but one. Conservation of energy is given by the time component, while conservation of momentum is given by the three space components, of one same space-time tensor law of conservation.

In a general way, all problems of dynamics or kinematics in three-dimensional space can be treated as problems of statics in four-dimensional space-time. All these advantages of space-time appeal very strongly to the theoretical investigator who deals with mathematical equations; but they all go to show how much simpler it is to understand the workings of the universe when we realise that the fundamental continuum of the world is space-time, and not separate space and time.

We must now mention another most important conception of classical science, namely, Action; and we shall see that this important mathematical entity finds its place in a perfectly natural way in the world of space-time. In order to understand the significance of Action, let us consider any mechanical system passing from an initial configuration to a final configuration . Classical science defined the action of this system as the difference between its total kinetic energy T and its total potential energy , taken at every instant and then summated over the entire period of time during which the system passed from the initial state to the final state .[122]

Now the total kinetic and potential energies of the system at any instant are given by where and represent the densities of the kinetic and potential energies at every point throughout the space occupied by the system. Accordingly, the expression of the action will be given by In the second formula we have merely replaced () by a single letter . Henceforth () or will be referred to as the function of action.[123]

Roughly speaking, action was thus in the nature of the product of a duration by an energy contained in a volume of space. On no account may this action be confused with the action dealt with in Newton’s law of action and reaction, also expressible as the principle of the conservation of momentum. Still less may it be confused with the term “action” which appears in philosophical writings. The importance of the conception of action arises from the fact that the laws of mechanics can be expressed in a highly condensed form when the conception of action is taken into account. We thus obtain what is known as the Principle of Least Action, with which the names of Fermat, Maupertius and Lagrange are associated. A more convenient form of the principle was given by Hamilton; it then becomes Hamilton’s Principle of Stationary Action. If we restrict our attention to the very simplest case, we may state Hamilton’s principle as follows:

If we consider all the varied paths along which a conservative system may be guided, so that it will pass in a given time from a definite initial configuration to a definite configuration , we shall find that the course the system actually follows, of its own accord, is always such that along it the action is a minimum (or a maximum).

Now the principle of action issues, as we have said, from the laws of classical mechanics. Were these laws different, the principle of action would cease to apply, at least in its classical form. A priori, we have no means of deciding whether the laws governing physical phenomena of a non-mechanical nature—those of electromagnetics, for example—would also issue in the same principle of action.

But when Maxwell had proved that his equations of electromagnetics could be thrown into a form compatible with the principle of action, and when he had succeeded in amalgamating electricity, magnetism and optics into one science, the universal validity of the principle was accepted. Inasmuch as this principle includes that of the conservation of energy, we can understand why the principle of action was often referred to as the supreme principle of physical science. Incidentally, we may mention that when the principle of action is satisfied by a phenomenon, an indefinite number of different mechanical interpretations of the phenomenon are theoretically possible. In the case of electrodynamic phenomena, however, in view of the complicated hypotheses which he was compelled to postulate, Maxwell abandoned all attempts to discover the precise mechanical interpretation which would correspond to reality.

Thus far, we have limited ourselves to stating the general validity of the principle of action, as applying to all physical processes. It now remains to be seen how the principle will be of use to us in furthering our knowledge of the laws of nature. This we may understand as follows: The principle, as we have seen, imposes the condition that the natural evolution of any system must be such as to render the action a maximum or a minimum. Could we but express this condition in terms of the usual physical magnitudes, we should be enabled to map out in advance the series of intermediary states through which the phenomenon would pass. From this knowledge we should derive the expression of the laws which governed the evolution of the phenomenon. Here, of course, a twofold problem presents itself. First, we must succeed in finding the correct mathematical expression for the action; and, secondly, we must be in a position to solve the purely mathematical problem of determining under what conditions the action will be a maximum or a minimum.

Now all problems of maxima and minima are solved by means of the calculus of variations, a form of calculus we owe chiefly to Lagrange. According to the methods of this calculus, we establish under what conditions a magnitude is a maximum or a minimum by discovering under what conditions it will be stationary.

Let us explain what is meant by the word “stationary.” When a stone is thrown into the air, it ascends with decreasing speed, then seems to hesitate for a brief period of time as it hovers near the point of maximum height before it starts to fall back again towards the earth. During this brief period of hesitation at the apex of its trajectory, the stone is said to remain “stationary.” We can recognise a stationary state by observing that when it is reached no perceptible changes take place over a short period of time. In this way, we may understand the connection which exists between the stationary condition and the presence of a maximum or a minimum. In mathematics small variations are represented by the letter ; hence the stationary condition of the action, or again, the principle of action, is expressed by The calculus of variations enables us to solve such purely mathematical problems. Under the circumstances the problem of determining the laws which govern any particular phenomenon reduces to discovering the expression of the function of action pertaining to the phenomenon in question. Larmor applied this method to the phenomena of electricity and magnetism and showed how Maxwell’s laws of electrodynamics could be deduced from a suitable mathematical expression defining the electromagnetic function of action.

When the theory of relativity supplanted classical science, it was recognised that the classical equations of mechanics were only approximate, and it became necessary to reformulate the principle of action so as to render it compatible with the mechanics of relativity, hence also with space-time. This work was carried out by the pure mathematicians—by Klein and Hilbert in particular. It was then found that a principle of action differing but slightly from the classical one could be obtained. But inasmuch as the difference involved presents only a theoretical interest, we shall discuss the bearing of the principle of action on the theory of relativity without taking the slight aforementioned modification into consideration.

In classical science, it was strange to find that action, though so important in its physical significance, should yet present the artificial aspect of an energy in space multiplied by a duration. As soon, however, as we realise that the fundamental continuum of the universe is one of space-time and not one of separate space and time, the reason for the importance of the seemingly artificial combination of space with time in the expression for the action receives a very simple explanation. Henceforth, action is no longer energy in a volume of space multiplied by a duration; it is simply energy in a volume of the world, that is to say, in a volume of four-dimensional space-time. Designating a volume of space-time by , we have so that our principle of action, , becomes There is now perfect symmetry between the rôles of space and time.

Now there was still another aspect of the principle of action which was extremely displeasing in classical science: By making the present course of a phenomenon contingent on a condition which could be determined only at some future date, it acquired a teleological character. But, as Planck points out, with the action as given by an energy in a volume of space-time, we may, by placing space and time on the same footing, regard duration as static; and the unsatisfactory teleological aspect of the principle may be obviated.

The space-time theory offers yet further advantages. It suggests that the function of action must be a space-time invariant. If, then, we know the elementary physical magnitudes which must enter into this invariant, purely formal mathematical considerations enable us to construct all possible invariants, and one of these will represent the function of action. Assuming the world to be built up of electricity and metrical field, the possibility of obtaining the “function of action” of the universe would seem to be within our grasp. Could this result be accomplished, a momentous discovery would have been made, since all the laws of the physical world would be comprised in the expression The application of the principle of action, expressed in the preceding formula, would enable us to derive all the laws of the physical universe. We should understand many things that are still mysterious; the elusive reason for the existence of two types of electricity in nature would be revealed, and the strange tendency of matter and energy to congregate into atoms and quanta would be explained. Unfortunately we are far from being able to determine the “function of action” of the universe, for there may be physical elements entering into its constitution whose nature we as yet ignore. (Furthermore, we might mention that the very validity of the principle of action is called into question by the quantum phenomena.) The following passage quoted from Weyl gives an idea of what is at stake:

“Whereas, in mechanics, however, a definite function of action corresponds to every given mechanical system and has to be deduced from the constitution of the system, we are here concerned with a single system, the world. This is where the real problem of matter takes its beginning: we have to determine the ‘function of action,’ the world-function , belonging to the world. For the present it leaves us in perplexity. If we choose an arbitrary , we get a ‘possible’ world governed by this function of action, which will be perfectly intelligible to us—more so than the actual world—provided that our mathematical analysis does not fail us. We are, of course, then concerned in discovering the only existing world, the real world for us. Judging from what we know of physical laws, we may expect the which belongs to it to be distinguished by having simple mathematical properties. Physics, this time as a physics of fields, is again pursuing the object of reducing the totality of natural phenomena to a single physical law: it was believed that this goal was almost within reach once before when Newton’s Principia, founded on the physics of mechanical point-masses, was celebrating its triumphs. But the treasures of knowledge are not like ripe fruits that may be plucked from a tree.”[124]

Whatever may be the ultimate fate of such speculations in revealing to us the mystery of matter, it cannot be denied that they possess a grandeur which is compelling. At all events, in more restricted domains, the principle of action has led to important discoveries.

Hilbert, Klein, Weyl and later Einstein himself, by applying the principle of action to the space-time metrical field, were led to the law of space-time curvature in the interior of matter and outside matter; that is to say, to Einstein’s law of gravitation.

The problem resolves itself into the selection of an appropriate expression for the action; but in the case of the law of gravitation a peculiar difficulty is encountered. We shall best understand the nature of this difficulty by considering the simpler case of the electromagnetic action. Electricity and magnetism present themselves in a substantial form as electrons, magnets and electrons in motion, or currents; and in a more ethereal form as fields, illustrated by the electromagnetic field. Inasmuch as both these manifestations of electricity and magnetism should enter into the laws of electrodynamics, the total electromagnetic action will have to comprise the two corresponding types of action. These can easily be determined, and an application of Hamilton’s principle of stationary action yields the laws of electromagnetics without further ado. But when we consider matter, classical science knew of but one form, the substantial form; the field form was lacking. To be sure, the gravitational forces surrounding matter appeared to present a certain analogy with the electromagnetic fields surrounding electrons and magnets. But, mathematically, at least, the analogy was superficial, for in contradistinction to the electromagnetic field, the gravitational field was not expressed by differential equations indicating continuous action through a medium. Only in name was it a field; and it appeared impossible to determine its action. With Einstein’s discoveries this difficulty was overcome. The gravitational field now appeared as a field analogous to the electromagnetic field. It was the metrical field of space-time, defined, as we know, by the potential distribution analogous to the potential distribution of the electromagnetic field. It can also be regarded as representative of the structure, or geometry, of space-time. Henceforth, the total material or gravitational action will be given by the action of matter proper plus that of the metrical field; the missing action has been found.

Thus the solution of the problem of gravitation reduces to establishing the mathematical expression of these two separate actions. Now the action of matter in a space-time region is the energy of matter contained in a volume of space-time. According to Einstein’s theory, therefore, it is given by the density of the mass of the matter multiplied by a space-time volume. As for the action of the gravitational field, inasmuch as it is now seen to be the action of the metrical field, it can be discovered without much difficulty. In this connection, we must recall that the relativity theory suggests that the action in a volume of space-time must be an invariant; its numerical value, whatever it may be, must be independent of our choice of mesh-system. In other words, the action of the metrical field must be an invariant built up from the fundamental tensors , whose distribution defines the metrical field.

We mentioned, when discussing tensors, that several such invariants exist, but that the simplest of all was represented by the Gaussian curvature . To this we may add any scalar or number without modifying its invariance, since numbers are, of course, invariants. Hence, the simplest expression for the function of action of the metrical field is obtained by writing it: . This was, indeed, the expression selected by Einstein.[125]

From now on, it is a mere question of mathematics (calculus of variations) to express the stationary condition of the total action. As a result, we obtain Einstein’s law of space-time curvature, inside and outside of matter; that is to say, Einstein’s law of gravitation. According to whether we assume that is zero or has a non-vanishing value, we obtain the infinite universe, or the finite universe of either de Sitter or Einstein. In particular, Einstein’s universe will be the outcome if we impose the condition of stability upon the universe. We see, therefore, that the principle of action proves that all three types of universe are possible. If we wish to decide which type corresponds to the real universe, we must take other facts into consideration, as was explained in the chapter on the finiteness of the universe.

Now there remains one other important point which we must mention before abandoning the discussion of the law of gravitation. We saw that the principle of action alone did not enable us to decide whether the constant , which was adjoined to the action of the metrical field, was zero or had some non-vanishing value. In other words, it did not enable us to decide whether the universe was infinite or finite. But this is not the only source of indeterminateness. We mentioned that Einstein chose for the function of action of the metrical field because it was the simplest possible invariant made up of ’s; but we have also seen that other more complicated types of invariants could be constructed. Had these been substituted in the expression of the action, we should have obtained alternative laws of gravitation. But the point we wish to stress is the following:

Regardless of the precise invariant we may select in order to represent the function of action of the metrical field, there is one property which is common to all these invariants. Their mathematical nature is such that when we apply to them the principle of action, we find that the curvature of space-time which would be present in the interior of matter must be of a type which would remain conserved. Granting that a certain indeterminateness may exist in our choice of the action, and hence of the law of gravitation, all these laws would compel us to recognise that in our universe, permanent entities such as matter cannot help but be present. To be sure, the type of conservation necessitated by the principle of action is not altogether the classical conception of conservation; but, as we saw in a previous chapter, the new principle becomes that of classical conservation when a Galilean reference system is adhered to.

Finally, we may note that by applying the principle of action to the total action formed by the gravitational action and the field-action of electricity, we should obtain the precise type of space-time curvature which would exist in an electromagnetic field free of electrons and magnets; for instance, in an empty region flooded with light. It is then found (as was mentioned in Chapter XXXIII) that the electromagnetic field also produces a curvature of space-time, less pronounced than that reigning in the interior of matter, but more pronounced than that existing in the empty space around matter (that is to say, in a gravitational field due to ordinary matter).