CHAPTER XXTHE IRREVERSIBILITY OF TIME
WE saw, when discussing the existence of a finite invariant velocity in nature, that the presence of this velocity was going to work havoc with our belief in absolute simultaneity. Thus, if two events take place in different places, we can no longer attribute any universal meaning to the opinion that these two events have taken place at one and the same instant of time or at two different instants. In other words, there exists no absolute clock giving us the correct time at all points of the universe. According to the motion of the observer, two events which are simultaneous for one may follow in sequence for another.
At this point we must guard against possible confusion. It might be argued: “There is nothing very new in this relativity of simultaneity and of the order of succession of our perceptions; for we encounter such examples even in classical science. For instance, if two explosions occur at different spots, by adjusting our position as observers it may be possible for us to perceive the two noises now simultaneously and now in succession.” But such arguments would indicate that the revolutionary notion of the relativity of simultaneity had not been grasped.
Thus, in the example of the two explosions happening in different places on the earth’s surface, while it is perfectly true that the order of succession in which we hear or see these explosions will depend on our position as observers, yet, on the other hand, if we take into consideration the time which sound or light waves require to cover the distances separating us from the two explosions, we can always decide without ambiguity whether the two explosions occurred simultaneously or in succession. It was never the order or the simultaneity of our perceptions which was considered absolute in classical science; it was solely the simultaneity or order in which these explosions or events had taken place in the outside world.
Now, according to the theory of relativity, not only is the simultaneity of our perceptions relative, but in addition, even when we take into consideration the speed with which the sound waves or light waves advance towards us, it is still impossible to decide whether the two events (regarded as existing in the outside world independently of any observer) are simultaneous or not; for our calculations would show that this simultaneity in the outside world would vary from one observer to another.
The relativity of simultaneity leads us to the kindred subject of the relativity of the order of succession of two events occurring in different places. Here it is impossible to lay down a rigid rule; for we shall see that the theory compels us to recognise that in certain cases the order of succession is indeterminate or relative to our motion, while in others it is absolute, remaining the same for all observers.
The problem is of sufficient importance to warrant a more detailed explanation. Our awareness of the passage of time constitutes one of the most fundamental facts of consciousness. Not only is time continuously passing, but it is ever flowing in the same direction. To this mystery of the uni-directional passage of time, the theory of relativity contributes no new information, so that we may discuss the problem from the standpoint of classical science. Suppose, then, that in a bottle we place two layers of powder, one white and one black. If we shake the mixture long enough, we know that the final result will be a uniform grey mixture. Now we might have anticipated this result without actually performing the experiment. For if we consider the various ways in which the particles of the powders could be distributed in the bottle, sufficient reason would urge us to assert that all distributions were equally probable. But it so happens that by far the greater number of these possible schemes of distribution would yield the appearance of a uniform grey mixture. Probability would suggest, therefore, that on shaking the bottle long enough, the uniform grey mixture would finally appear. When this homogeneous state had been obtained, only once in æons of time would the black and white separation reappear, and then but for an instant. We may express these facts by saying that the general tendency would be a passage from the heterogeneous (black and white layers) to the homogeneous (uniform grey mixture).
The example we have considered is one of extreme simplicity, but inasmuch as the arguments involved appear to be of universal validity, we may say that natural phenomena present a uni-directional sense of advance, passing from the states of lesser probability to those of greater probability. We may therefore identify the states of lesser probability with the past and those of greater probability with the future. The direction of time’s passage can thus be defined physically in terms of probability considerations, entailing a mere appeal to sufficient reason and to operations of counting.
This was the definition suggested by Boltzmann (the founder of the kinetic theory of gases). According to him the universe was passing from states of lesser probability to that of maximum probability, and the direction of this passage defined that of time. As can be gathered from the preceding explanations, a uni-directional passage in the course of phenomena is to be ascribed to the fact that phenomena are irreversible,i.e., that the various states are not equally probable.
It is true that there also exist types of phenomena for which all states present the same probability. In this case no privileged direction exists, and the phenomena are calledreversible. An adiabatic transformation under ideal conditions, and the rotation of a body in the absence of friction, are illustrations of reversible processes; such phenomena would of course be incapable of defining the direction of time.
But by far the greater number of phenomena in nature are of the irreversible type; with these a definite direction of change is privileged. When we wish to force the phenomenon against its natural trend, work must be furnished. All phenomena entailing friction are of the irreversible sort, for whereas motion generates heat through friction, the heat cannot be used to regenerate the motion. The example of the two powders also presented us with an illustration of the irreversible type of phenomenon, since the natural evolution was from heterogeneity to homogeneity. To be sure, it would be possible to reverse the process, but only through the medium of some intelligent activity sorting out the particles and distributing them according to states of lesser probability. The action of a demon of this sort, Maxwell’s demon, would cause the direction of the irreversible phenomena to be reversed, so that the direction of time would appear to change. Needless to say, however, Maxwell’s demon is but a fiction.
In the illustration of the two powders we can readily understand the reason for the irreversibility, but it was not by this method that the principle involved was first discovered. We must go back to Carnot and to his investigations on the cycle of the steam engine in order to trace the origin of what was to become one of the most fundamental principles of science. Carnot’s celebrated principle relating to the efficiency of the steam engine was shown by Clausius to be a special case of a general physical principle which may be stated thus: “Heat cannot flow unaided from a colder to a hotter body, but tends invariably to seek lower levels of temperature.”
By introducing a new concept calledEntropy, defined as the ratio of a quantity of heat to a temperature, Clausius was able to give a more general form to this principle. It then became thePrinciple of Entropy, according to which, in any irreversible change, the entropy of a system was increased; only in the case of a reversible change would it remain constant. Under no circumstances, however, would the total entropy decrease. Thus all natural processes involve an increase of entropy since none are ideally reversible.
Now when we consider the principle of entropy in its bearing on the concept of energy, we find that it implies a continual loss of potentiality in the energy of the system. The energy is thus constantly dissipated, or, more properly speaking,degradedinto that lowest of forms, namely, disorderly molecular agitation,i.e., heat; whence the name,Principle of the Degradation of Energy, under which it is often referred to.
The principle is of such extreme importance that it has been named theSecond Principle of Thermodynamics; the appellationFirst Principle of Thermodynamicsbeing reserved for the principle of the conservation of energy. It is important to note that these two principles, that of degradation and that of conservation, do not conflict. In quantity the energy is conserved; it is solely in quality, or in potentiality, or in its ability to perform work, that it is degraded.
We now come to Boltzmann’s contributions. Entropy as defined by Clausius was an exceedingly abstruse thermodynamical concept. Boltzmann, by basing his deductions on the kinetic theory of gases, succeeded in giving a more concrete representation of entropy, defining it as the logarithm of a probability. Thus, if we admit the correctness of the kinetic theory, the principle of entropy becomes one of maximum probability, losing thereby its status of absolute validity and assuming a statistical significance. The representation of entropy in terms of probability permits an easy application of the principle to the problems of molecular physics and to atomistic physics in general. We may note that it was by following this method, or rather by combining the two definitions of entropy, that Planck established his quantum radiation formula, and was thus led to his quantum theory. Needless to say, the principle holds only when exceedingly large numbers are considered. In other words, it is a principle governing the chaos, or, again, it is a statistical principle.
It will be seen that the principle of entropy, by stating that the world is ever passing from states of lesser to states of greater probability, indicates that when the state of maximum probability or entropy is reached, no further change can take place. There will then be silent immobility or death. Under the circumstances it might appear that the principle was inconsistent with the existence of an original state of minimum probability. But here it should be remembered that the principles of science do not aspire to any absolute measure of value. All we can demand is that they be satisfied in the very restricted domain which science can explore. In the case of the principle of entropy, however, there is no reason to fear any conflict with the possibility of rebirth or re-creation. For instance, when shaking the powders, we saw that a uniform grey mixture would be the outcome; but were we to go on shaking the bottle for trillions of centuries, at some time or other the black and white separation would reoccur, and the cycle begin all over again. In short, the entropy would continually increase, but once every trillion years it might suddenly decrease, and then start to increase once more. We may mention that although there is no reason to assume that vital phenomena are in any wise inconsistent with the principle of entropy, it is safer to restrict the principle to the physical world. With biological phenomena, questions of probability are far too complex to be treated with any measure of assurance in the present state of our knowledge. Only in certain special cases, such as those connected with Mendel’s observations, have probability considerations been introduced with any profit.[70]
Now the two points we wish to stress are as follows: First, the uni-directional progress of time is imposed by common experience. Secondly, the principle of entropy is suggested both by experience and by arguments of a rational order. We do not assert thereby that the principle of entropy, even when reduced to the rank of a statistical principle, can be claimed with assurance to be of universal validity. It is far from impossible that in certain phenomena, of which as yet nothing is known, some counterbalancing activity may be at work. Nevertheless, it may be pointed out that a denial of the principle of entropy, at least as regulating those physical phenomena with which science deals, would arrest further progress.
For instance, we can readily understand in what an embarrassing situation we should be placed were a reversal of time to occur. Quite aside from such trivial examples as that of a hen becoming a chick and disappearing into an egg, or from effects becoming causes, a number of more scientific considerations must be mentioned. Phenomena would now pass from states of maximum probability to those of minimum probability; we could not merely interchange the meanings of the words “maximum” and “minimum,” for these are given by purely numerical considerations, which are quite irrelevant to the direction of the time-flow. Hence phenomena would pass from the homogeneous to the heterogeneous. But whereas a homogeneous state is unique, and can be predetermined, a heterogeneous one is more or less arbitrary, and baffles prevision. Thus, in the case of the two powders, the homogeneous state is a uniform grey mixture; whereas a heterogeneous state might be a succession of alternate layers, or only two layers, or a juxtaposition of black and white cubes. In fine, prevision would become well-nigh impossible.
From this we may readily anticipate that should Einstein’s theory render the direction of time’s progress indeterminate, should past and future be interchangeable according to our circumstances of motion, the utmost chaos would ensue. However, we need have no fear on this score, for as we shall see, the theory of relativity leads to no reversal of causality. Let us now consider the space-time theory in greater detail.
When we consider the four-dimensional space-time continuum, where space and time are on the same footing, there is nothing to suggest either a flowing of time or a privileged direction for this flow. In order to conform the theory to the facts of experience, it is therefore necessary to postulate that our consciousness rises along the world-line of our body through space-time, discovering the events on its course. Obviously, we might reverse the presentation by assuming that it was space-time that was moving past our consciousness; we might also claim that the very texture of space-time possessed dynamic properties urging our consciousness along time directions. But in view of the vagueness of the subject, not much is to be gained by speculating on this score.
Now when, along the world-lines which each one of us follows, a common direction from past to future has been specified, we find that an absolute past and an absolute future (though not an absolute present) are demanded by the theory of relativity. To be more explicit, we find that whereas, according to the nature of our relative motions, certain events may appear as simultaneous, or as antecedent and consequent, or as consequent and antecedent, others will invariably present themselves in the same order of temporal sequence regardless of our motion.
This division between events whose order of occurrence is absolute and those whose order is relative is obtained as follows: Whenever it is possible for a ray of light to pass from one event to a second event, leaving the first event at the instant it is produced and reaching the second event before or even at the instant at which it is produced, the time sequence of the two events cannot be reversed by relative motion; it is absolute.
On the other hand, if it is impossible for the ray of light to satisfy the preceding requirements, owing to the too great distance or the too small period of time separating the two events, the time sequence of the two events becomes indeterminate. One observer may claim thatprecedes,whereas the other may claim thatprecedes,and yet another may assert thatandare simultaneous. If we remember that in Einstein’s theory no active influence, no propagation, can be transmitted with a speed greater than light, we see that when a ray of light cannot proceed fromto,leavingwhenis produced and reachingwhen or beforeis produced, it is quite impossible for us to suppose thathas influenced.No causal connection can exist between these two events. Hence we see that when two events are causally connected, their order of occurrence will remain the same for all observers. Only when no causal connections could possibly exist can the temporal sequence of the events be reversed by an appropriate motion of the observer.
It is not because a causal connection happens to exist between two events that the order is non-reversible; the non-reversibility is due entirely to the spatio-temporal separation of the events. As for the causal connection, it may or may not exist. The only reason we mention causal connections with reference to this problem is in order to show that there is nothing in the relativity theory to suggest a reversal of causality; hence, though Einstein’s theory entails the relativity of simultaneity and in certain cases that of temporal sequence, there is no danger of a cause appearing as an effect or vice versa. In particular, no relative motion of the observer could ever lead him to believe that the glass we had let fall had in reality leaped up from the floor into our hand. No danger of the black and white powders which we had shaken into a grey mixture, appearing to separate back into black and white, as a result of a change in our relative motion. In short, no danger of the principle of entropy being overthrown.
We may also add that whereas, in classical science, owing to the possible existence of influences transmitted with infinite velocity, any two events happening in space might have been conceived of as causally connected or at least as related, Einstein’s theory, by requiring the velocity of light to be unsurpassable, permits us to restrict the cases of possible causal relationships and thereby to gain a better understanding of the cross influences which may be active in the universe.