TOWARDS A UNIFIED COSMOLOGY

by     Reginald O. Kapp

PART IV - GRAVITATION

Chapter 21 - Inert, Gravitational and Attracting Mass


21.1: Conceptual Distinctions
It has been pointed out by others that the word 'mass' has three distinct meanings. Arthur Zinzen, for instance, discussing these in his Praktische Naturphilosophie (1953, Westkulturverlag), quotes G. Hamel, who had previously distinguished between these meanings in 1927 (Hand- buch der Physik, Vol. 5, Julius Springer). The three meanings are named by him as follows:

(1) Inert mass
(2) Attracted (weight) mass
(3) Attracting mass

The first two of these meanings have also been discussed by Einstein in his The Meaning of Relativity. He used the respective words 'inert' and 'gravitational' mass. I shall use the same ones here, but Einstein had no need to discuss the third meaning and so he gave no name to it. I shall therefore use Hamel's name 'attracting mass'.

Before its significance is discussed it has to be made clear why mass in its first two meanings is important in general relativity. For this purpose it is best to use Einstein's own, very precise, words. On page 55 of The Meaning of Relativity, Sixth Edition, he said:

'The ratio of the masses of two bodies is defined in mechanics in two ways which differ from each other fundamentally; in the first place, as the reciprocal ratio of the accelerations which the same motive force imparts to them (inert mass), and in the second place, as the ratio of the forces which act upon them in the same gravitational field (gravitational mass).'

In this passage the inert mass of a body is said to be the property the value of which determines the acceleration of the body under the influence of a given applied force. The gravitational mass, on the other hand, is said to be the property of the same body the value of which determines its weight in a gravitational field. The inert mass of a billiard ball, Einstein would have said, causes the ball to receive a finite acceleration when propelled by a billiard cue over a smooth, level surface. In the absence of its inert mass the acceleration would be infinite. The gravitational mass causes the same ball to deflect a spring balance. In the absence of gravitational mass the ball would not be attracted by the earth. The distinction between these two kinds of mass is made even more precise in a later passage:

'Newton's equation of motion in a gravitational field, written out in full, is: (Inert mass). (Accelerafion) = (lntensity of the gravitational field). (Gravitational mass).'

The left hand side of the above equation defines the performance of a billiard ball when a force, F, is applied to it by a billiard cue. If the inert mass is m1 and the acceleration f1, one can write:

F = m1f         ......         (21a)

The right hand side defines the weight of the ball on a spring balance. Let this weight be W and the gravitational mass mg The intensity of the gravitational field near the earth is g, so the algebraic expression for the right hand side is

W = gmg         ......         (21b)

Equations (21a) and (21b) are found to give numerically identical results for the same ball when f = g. From this Einstein inferred that mg = mi. He spoke of 'the law of the equality between the inert and the gravitational mass' and made it clear that 'the numerical equality is reduced to an equality of the real nature of the two concepts'.

General relativity is based on this identity, so that Einstein could say:

'The possibility of explaining the numerical equality of inertia and gravitation by the unity of their nature gives to the general theory of relativity, according to my conviction, such superiority over the conceptions of classical mechanics, that all the difficulties encountered in development must be considered as small in comparison with this progress.'

This is very clear and, I think, irrefutable. We observe that the ratio mg / mi is always a constant and ask why. Einstein's answer is that, if general relativity is accepted, they are one and the same thing.

Zinzen, in the book referred to above, speaks, nevertheless, of 'a really great confusion of thought' about the nature of mass, and I have been led to the reluctant conclusion that he is right. For I have found very little appreciation of the distinction between attracting mass and the other two kinds. If the distinction is hardly ever mentioned, it might be because it is too obvious to need mention, and I thought at one time that it was so. In an earlier draft of this book I therefore took the distinction for granted, But comments from sundry authorities who saw the draft convinced me that the true reason why the distinction is not mentioned is that it is not even thought about. I was told, for instance, that the word 'mass' can never; have any other but the two meanings that figure in the above quotations from Einstein; that attracting mass and gravitational mass are known by everyone to be identical both numerically and in their nature; that this identity is proved by Newton's third law of motion, according to which action and reaction are equal and opposite; that general relativity does not rest on the identity of the first two kinds of mass only, but on that of all three kinds; that conceptual distinctions are hair-splitting and fruitless. All these statements are, I think, erroneous.

If the distinction between the three kinds of mass and its importance are clear to many, they are evidently not as universally known as they need to be. I have therefore no choice but to draw attention to some rather elementary facts about mass and gravitation before I can proceed further with the present inquiry. One of those unfortunate situations arises here that do occasionally arise in science when an obstacle to research and discussion is not raised by the difficulty of the subject but by an all-too-prevalent notion that the subject does not present any problem at all. Here the problem to be discussed is why the ratio of attracting mass to the other two kinds is observed to have a constant value. One cannot begin to answer this question until one has achieved an understanding of the nature of attracting mass, and in view of the widespread confusion that I have encountered about it I make no apology for presenting some elementary facts here.

Equation (21 a) represents the performance of a billiard ball in one kind of circumstance, namely, when it is being propelled by a cue. Equation (21b) represents its performance in another kind of circumstance, namely, when it is being weighed on a spring balance. In the first instance the performance is accelerated motion, in the second, deflection of the spring.

Einstein did not need to ask questions about the circumstances. He did not ask, for instance, how a push came to be given to the billiard cue. The question would be as irrelevant to his theme as the names of the players. Nor did he ask what was the source of the gravitational field in which the ball is weighed. It sufficed for his purpose to note that there was a field. He needed to speak of its intensity only, not of its source or cause. But it is now necessary for us to turn our attention to the source.

The intensity of the gravitational field has the letter symbol g. If M is the mass of the earth, one can write

g = GM / x2

where G is the gravitational constant and x the distance between the centres of gravity of the earth and the ball. The question now is what suffix to give to M. The mass is the property of the earth by virtue of which it attracts the billiard ball. It is what Hamel called the 'attracting mass'. So I shall use suffix a. For the case when f = g Newton's law can then be expressed algebraically in such a way that all the three kinds of mass are distinguished by their appropriate suffixes as follows:

F = mi g = W = ( G Ma / x2 ) mg         ......         (21c)

21.2: The Behaviour of Masses in Newtonian Mechanics
Equation (21c) expresses the behaviour of masses in Newtonian mechanics when appropriate suffixes are used to distinguish between the three meanings of mass. Translated into words the equation yields three important statements in each of which the word mass has a different one of its three meanings. They are:

(1) The velocity of a body having inert mass is changed only when a force is applied to it. The relation between the force and the acceleration is as given by equation (21a), which shows that, for a given acceleration, the force is proportional to the inert mass, mi of the body.

(2) A body having gravitational mass experiences a force when it is in a gravitational field. The force is proportional to the gravitational mass of the body, mg , and the intensity of the field, as shown in equation (21b).

(3) A body having attracting mass is the origin of a gravitational field, the intensity of the field being proportional to the attracting mass, ma.

Be it remarked in passing that Newton did not use the word 'field', but this word is today a suitable one for making statements in Newtonian mechanics.

Here is a definition of inertia: The property of a body by virtue of which it resists a change in its velocity. Hence statement (1) is true by definition. The measure used to express the magnitude of the inertia is, in Newtonian mechanics, the force needed to procure unit acceleration. The formula (21a) can thus be used as a means of defining this force. It expresses the definition in quantitative terms. Statement (1) is a means of recognizing inertia when one meets it and does not do more than define inert mass. To make the first statement is but a way of saying that inert mass is inert mass. Like many useful statements in physics it is a tautology.

If one defines gravitational mass as the property by virtue of which a body experiences a force in a gravitational field, the second statement is also a tautology. It is true by definition of gravitational mass. But it is not a tautology to say that inert and gravitational mass are always equal. In Newtonian mechanics this statement appears as a necessary clause in the Cosmic Statute Book. It was Einstein who showed that the equality need not appear there as it can be inferred from the more basic principle of general relativity. This was a great step towards the unification of physics.

The third statement is also a tautology in so far as it serves to define attracting mass. Again, what is not a tautology is the statement that there is a constant ratio between the attracting mass of a body and its other two masses. This is the fact that can at present only be gleaned from observation. It is not implicit in the definitions of any of the three kinds of mass. The definition of inert mass tells us that a body possessing inert mass will be accelerated if it is pushed or pulled by something. The definition does not tell us that the mass will itself do any pushing or pulling. The definition of gravitational mass tells us how a body possessing this kind of mass is influenced by its environment. The definition does not tell us how a body having gravitational mass influences its environment.

The distinction can be expressed in a different way. Inert and gravitational mass can be measured only by observing the body that possesses them and in order to make the measurement something has to be done to this body. It has to be pushed or pulled in order to measure the inert mass and it has to be placed in a gravitational field in order to measure the gravitational mass. Attracting mass on the other hand can be measured only by observing its effect on a different body and in order to make the measurement something has to be done to this other body. Thus the inert mass of a train could be measured by doing something to the train itself, namely by accelerating it and measuring the tension in the draw-bar. The gravitational mass of the train could be measured by placing it on a weighing device. But one could not measure the attracting mass of the train by any observation on the train itself. One could, however, measure this kind of mass, in theory though not in practice, by bringing a different body, a pendulum for instance, near the train and observing the effect of the train's attracting mass on the deflection of the pendulum. The gravitational mass is measured by the effect of the earth on the train, the attracting mass by the effect of the train on some other gravitational mass. (This other gravitational mass might be the earth, as will be shown later).

One could express the difference, if perhaps a little loosely, by saying that inert and gravitational mass are properties of the body while attracting mass is a capacity for affecting other bodies. Or one could say alternatively that inert and gravitational mass are observed while attracting mass can only be inferred. But how best to express the difference is not as important as to remember that there is a difference. Use of the one word 'mass' for things that are conceptually so distinct has had some unfortunate results.

21.3: Mutuality in Gravitation
The notion that the equality of action and reaction proves the identity of gravitational and attracting mass is based on a misapprehension about elementary mechanics. But as it seems to be rather prevalent it ought not to be ignored.

Let us return to equation (21c). It gives a numerically correct result; so it contains everything needed for weighing the billiard ball. But it fails to embody a statement that is to be found in elementary textbooks. This is that the billiard ball attracts the earth at the same time as the earth attracts the billiard ball. This statement attributes an attracting mass, ma to the ball, which does not appear in the equation. We must find out where and how to show it.

Mutuality implies that the ball attracts other objects with a faint gravitational field. So we can represent the field intensity of the ball as gm

We then have to write gM for the field intensity due to the earth. The intensity, gm is of course very small indeed and can be neglected for practical purposes. But when the ball is replaced by the moon it is no longer so. The moon does not have an orbit around the centre of the earth but around the common centre of gravity of both bodies, and this is because the moon has a significant field of its own. In computing their relative movement one gives an acceleration, gM to the moon toward the earth and another acceleration, gm of the earth towards the moon The case of the billiard ball is similar. When the billiard ball is free to move, it does so in the earth's field, gM, while the earth moves towards the ball in the field of the latter, gm. If one wants to be pedantically accurate one must therefore define the relative motion of the billiard ball and the earth as determined by

g = gM + gm         ......         (21d)

In this equation

gM = k Ma / x2

gm = k ma / x2

The pull of the earth on the ball is gM,mg , while the pull of the ball on the earth is gm,Mg. The very small field intensity due to the ball is compensated for by the large attracted mass. One can say that the heavy earth is being weighed in the faint field due to the ball at the same time when the light ball is being weighed in the strong field due to the earth. In these circumstances both show the same weight and contribute equally to the deflection of the spring. The ball presses downwards on the top end of the spring and the earth presses upwards on the bottom end. In producing a compression the forces are additive; hence the plus sign in equation (21d).

Let the pull of the earth on the billiard ball be Fe and the pull of the billiard ball on the earth Fb. One can then write

Fe = ( k Ma / x2 ) mg

Fb = ( k ma2
) Mg

These two forces are equal and opposite. Action and reaction are also equal and opposite. Therefore, it has been argued to my surprise, these forces are action and reaction. But are we, according to this argument, to regard Fe as action and Fb as reaction, or vice versa? The difficulty of saying which is which ought to serve as a warning against so careless a conclusion.

The true situation can be understood with the help of the diagram in Fig. 4. Fe is shown as acting on the top of the spring and has its own equal and opposite reaction. The same holds for Fb, which is shown as acting on the bottom of the spring. Each force could equallv well be shown in anv other place between the ball and the earth. It is only for convenience of presentation that they are shown separated. The important fact is that if either Fe, or Fb disappeared, the other force would still be there with its own reaction.

The reader may have difficulty in believing that the very weak field of the billiard ball has any effect at all on the spring. It will help him if he imagines that mass is being continually transferred from below the spring balance to the tray where the billiard ball rests. If this goes on until half the mass of the earth is on the tray and half left below the spring balance


Fig. 4. Action and Reaction produced by gravitation

the situation will be reached when half the earth is weighed in the field of the other half. A gramme mass will then have half a gramme weight. If the process continues until no more than a billiard ball is left beneath the spring balance practically the whole of the earth will be weighed in the field of this billiard ball. In this field a gramme mass will weigh very little. The compression of the spring will be the same as before the transfer of substance was begun.

From the above considerations it follows that the pedantically precise algebraic expression for the weight of the billiard ball is

W = ( 1/2 G Ma / x2 ) mg + ( 1/2 Gma / x2 ) Mg         ......         (21e)

from which it appears that k , above, equals 1/2G.

Equations (21e) and (21c) are numerically equal, but this is only because Ma mg = ma Mg. One cannot infer from this equality that attracting and gravitational mass are of the same nature.

Their numerical equality, it must also be noted, has been obtained by a suitable choice of G. If one had arbitrarily chosen G as unity, one would say that one unit of gravitational mass is always equal to G units of attracting mass.

The value of G has been determined by experiments with more or less massive spheres by Cavendish, Poynting, Boys and others. These experiments show that the spheres have attracting mass as well as inert and gravitational mass. But they do not prove that less massive bodies also have attracting mass. It is here that equation (21e) is valuable. If one were weigh a body that had gravitational mass only and no attracting mass, would have half the observed weight. But the smallest objects that give observable deflection on a spring balance have the weight that one would predict on the assumption that they have attracting mass. One may therefore safely conclude from observation and experiment that the law of constant proportionality between inert, gravitational and attracting mass holds for very small accumulations of matter.

It is, nevertheless, important to bear in mind that this mutuality in gravitation is based on observation and experiment and not derived from any more fundamental principle, at least in Newtonian mechanics. If one were to find a body that had no attracting mass, it would not be weightless. Action and reaction would apply to its pressure on a spring balance. Every other principle of which I am aware would also be preserved. I can think of no way of proving the impossibility of a body that. has no attracting mass having gravitational mass except by the inductive kind of reasoning that takes the line: This has never been observed; therefore it cannot happen.

Before I leave the discussion of the three kinds of mass in terms of Newtonian mechanics I should like to point out that one cannot discover whether a body does or does not possess attracting mass by observing it while it falls freely. One can do so only by weighing it. For the equation of the free fall is

gm1 = ( GMa / X2 ) mg         ......         . (21f)

The attracting mass, mag = gM + gm to a minute extent. A ball that had no attracting mass would fall with practically the same acceleration as an actual one One could only observe the lack of attracting mass in a falling body if the ratio gm / gM were large enough to be significant.

It should be remembered that it was from equation (2If) that Einstein developed general relativity. This equation established for him the identity in nature as well as numerically, of the inert and gravitational mass of mass m, namely mi and mg. And this equation gives no information about the attracting mass Ma. It is therefore impossible to reach any conclusion about the attracting mass of m so long as one uses equation (21f).

The question arises what part, if any, attracting mass plays in general relativity. Only experts can give the answer, but I should like to say that I have been able to find but little discussion of this question. This may, however, be due to my meagre reading. General relativity does explain quite clearly why a gravitational mass is accelerated when it finds itself in a gravitational field, but, so far as I can make out, it does not account for (and does not need to account for) the field. What is basic in general relativity seems to me to be equally true whatever the object may be that is the source and cause of the field, by whatever process the field is produced.

Some relativists to whom I have spoken about this have taken it for granted that in all relativity equations the symbol that represents gravitational mass automatically represents attracting mass as well. It is an assumption that calls for great caution. I am sure that the basic conclusion of general relativity does not depend on it. I do not know whether any subsidiary conclusions do. If so, I venture to suggest that they ought to be carefully scrutinized.

That general relativity does not depend basically on the identity of attracting and gravitational mass seems to me not only to follow from the equations but also to be apparent when one translates into the language of general relativity the three statements about mass that have been expressed above in the language of Newtonian mechanics. This will be done in a moment, but it is necessary to lead up to the translation by first explaining the nature of the gravitational field as it appears in general relativity.

21.4: Gravitation as Interpreted in Relativity Theory
In Newtonian mechanics a gravitational field appears as a region in which force acts at a distance from its source. Action at a distance had always been regarded as an unsatisfactory concept. One thinks of a force as applied by physical contact between something and ponderable matter, such as when a thing is pushed by a stick or pulled by a string. The notion could never be easily accepted that a force could be exerted at a distance from any physical object. Yet this is what seemed to happen to a stone that was attracted by the earth. To make this notion a little less unsatisfactory an hypothetical ether was postulated. It was assumed that a gravitational mass produced some kind of effect, called a strain, in the ether and that the force was applied by physical contact between the stone and this strained ether. To give a more concrete meaning to this assumption the strained ether was later said to contain tubes of force that extended between the attracting earth and the attracted stone. Thus every massive body was believed to have a twofold environment, to be surrounded at the same time by a featureless, undifferentiated space and by a featured ether.

It has been said that Einstein abolished the notion of a featured ether. One might equally well say that he abolished the notion of a featureless space. He retained the concept of a featured environment with physical properties but showed that there was no need to assume an additional featureless environment. For this single environment with physical properties he wisely retained the name space. But what he did abolish was the concept of space as the container of the material universe. Einstein's featured space is a constituent of it.

One of Einstein's great achievements was to define precisely the physical condition of a gravitational field. It had been previously vaguely thought of as a strain. He showed that this condition can be described as a departure from Euclidean geometry. Space that shows such a departure is technically called curved. Einstein further proved that the kind of motion that would appear to be at constant velocity in flat space would appear to be accelerated motion in curved space. If a stone moved out of nearly flat space into a gravitational field, as would happen if it fell from a very great height, its velocity would change, but that would not mean that it was subjected to a force. In other words, one must not think of the acceleration of a falling stone as the consequence of what is done to the stone but as the consequence of where it is, i.e. the acceleration is a function of the local geometry.

The new outlook made it necessary to define inertia more precisely than had been necessary with Newtonian mechanics. In relativity language inertia is the property by virtue of which the acceleration of a body that is free of all restraint depends on the curvature of the space in which it finds itself. If the space has zero curvature, the acceleration is zero. In this respect relativity is not a denial, but an extension, of Newtonian mechanics.

One can conceive a line in four-dimensional space-time that a freely moving body would trace when not subjected to a force. According to Newtonian mechanics it would always be a straight line and represent a constant velocity. According to relativity theory it would only be a straight line when the space was flat. When the space was curved, though the body was not subjected to a force, the line would be curved and represent an acceleration. Thus to the question: How can a falling stone be subjected to a force without physical contact with ponderable matter? Einstein's answer was as beautifully simple as it was unexpected. He said that the falling stone is not subjected to a force. A force is no more needed to maintain an acceleration in curved space than to maintain constant velocity in flat space. On the contrary a force would be needed to prevent acceleration in curved space. A stone is subjected to such a restraining force when it is at rest on a shelf, the force being exerted by physical contact with ponderable matter, namely the shelf. In flat space an observer who is being accelerated experiences a force. In curved space it would be an observer who was not being accelerated in the direction and at the rate defined by the curvature who would experience a force.

21.5: The Behaviour of Masses in Relativity Theory
Now let the three statements about mass that were formulated above in the language of Newtonian mechanics be translated into the language of relativity:

(1) The acceleration of a body having inert mass and not subjected to any force depends on the geometry of the space in which the body is. The acceleration is zero when the space is flat and has a finite value when the space is curved.' It would be idle to discuss whether this is a tautology. Whether it is or not it is a definition of inert mass in the language of relativity just as the equivalent statement given earlier here is a definition formulated in the language of Newtonian mechanics.

(2) 'A body that is free of restraint is accelerated when it is in the curved space that constitutes a gravitational field.' This is but to mention a special case of (1).

(3) 'A body having attracting mass is itself the cause of a curvature in space, the value of the curvature being proportional to the value of the mass.' This is no tautology. It is not implicit in the definition of inertia. There does not appear to be any reason why a body that experiences an acceleration when the space around it is curved should itself have any influence on the curvature of the space. No one has been able to show that the one fact is a logical consequence of the other. Though relativity has explained much, it has not explained the apparent invariable association of inert and attracting mass. One is still left with the question: Why has one never observed an inert mass that leaves the space around it undisturbed?

It has become clear that both Newtonian mechanics and general relativity leave us with the same puzzle about attracting mass. Observationally, this and the other two kinds of mass, inert and gravitational, are always coupled. But no reason has been given why it should be so. Why, one is led to ask, does one never observe a body that has inert and gravitational mass but has no attracting mass? Why, in relativity language, does one never observe a body with inert mass that leaves the space around it undisturbed?

I shall suggest a possible answer later on, but much ground remains to be covered before this can be done.

21.6: Two Observers or One?
After this book had been sent to the printer I came by accident upon two papers published in the Physikalische Zeitschrift of 1921. They are by one E. Reichenbacher. In these papers much is said that has been said in the present chapter. It is said very clearly and cogently, if in a different way. Reichenbacher urges many sound arguments against identifying the gravitational with what he calls, rather unfortunately, the inert field. At the same time he shows complete understanding of the reasoning that led Einstein to identify inert with gravitational mass. Reichenbacher distinguishes between passive and active gravitational mass. This is the distinction that has been made in the present chapter between gravitational and attracting mass.

It is strange indeed that this distinction has not received greater prominence in spite of the fact that attention was drawn to it with some insistence in the very early days of general relativity. In these two articles Reichenbacher is revealed as a clear and powerful thinker. To understand why his contribution was so ineffective one must recall the atmosphere in the early nineteen-twenties.

Concerning general relativity theory scientists found themselves in two opposing camps. Those in one camp rejected the whole of the theory, not without emotion. Those who did not express themselves violently on the subject did so peevishly. Scientists in the other camp supported relativity theory equally vehemently. Aware of the immense clarification and unification that it had achieved they saw themselves as crusaders for a new revelation. Loyalty to the cause, loyalty to Einstein impelled them. Those who did not understand relativity theory in the least rejected the whole of it. Those who did understand it, more or less, accepted it in its entirety, often uncritically. As defenders of the Faith they could tolerate no doubts. They reacted as we all do at times against the nagging suspicion that something needed further clarification.

In such an atmosphere Reichenbacher could hardly escape being unpopular with both camps. He understood general relativity so well that he could venture to explore its limitations. For him doubts about the identity of the inertial and the gravitational field were not nagging. He faced them squarely. Doing so he came to realize that the gravitational field can be described as a distortion of space such as to give it a non-Euclidean geometry and that this distortion is independent of an observer in a way that causes the gravitational field to differ from the 'field' that he saw as the result of accelerated motion and that he called one of acceleration. That Reichenbacher's analysis led him to a clearer explanation of centrifugal force in relativistic terms than has been provided by other relativists makes it all the more regrettable that his work should have received so little notice. Relativists of his time would seem to have disliked the awkward questions that he propounded all too inescapably. Unable to refute him thev ignored him.

I am bold enough to suggest that this reveals one of the less healthy aspects of the science of relativity. Today opportunities for sanitation are rarer than they were in the nineteen twenties. In those years many of the best intellects trained in physics devoted a great deal of their time and thought to relativity theory. Today experts in this field are but few. Special relativity occupies no more than a minute fraction of the physics syllabus in our universities and general relativity is hardly taught there at all. This is unfortunate. We are in danger of losing the great benefit, the considerable insight, that relativity theory brought. It has more to tell us, much more, than the formulae expressing the Lorentz transformation. I am hoping that the following chapters will illustrate how a revival of interest in general relativity can prove rewarding.

If different interpretations of general relativity were possible in 1921 they are certainly still possible today, for there is little, if anything, to show that the difficulties to which Reichenbacher referred have been resolved, or even much appreciated. There can be no change unless interest in general relativity becomes more widespread and it may encourage some to turn their attention to this subject if I give a very brief account of its basis. For this purpose the mathematical treatment is not relevant and I shall not attempt it. I shall follow as closely as possible the lines taken by Einstein and other brilliant expositors of the past, while departing from these lines just enough to show why those who ignored, and have continued to ignore, the difficulties have not been guided by understanding but by careless thinking.

In the early days of relativity we were introduced to the notion of a box in space, far distant from any ponderable matter. Einstein was the first to invent this illustration. A man was supposed to be inside the box and to make all the observations that could enable him to learn something about its motion through space. The man could secure himself to the floor of the box. He could climb to the ceiling and catch hold of a strap attached to it. He could let go of the strap and observe what happened. He could measure any forces that were exercised on his body. When he had observed and measured he could apply logic and mathematical reasoning to the results and arrive therefrom at some conclusions about the nature of space, time and gravitation.

It will be convenient to refer to the observing, reasoning man in the box by a name. I should call him Mr. Einstein were it not that I might then mislead the reader and also divert attention from the subject of relativity to the question what, in fact, Einstein did or did not say, what he did or did not think, what he ought or ought not to have said or thought. For what I have read and what I have learnt in conversation has shown that different people have different ideas about these matters. So the man inside the box shall be called Mr. Smith.

Let us begin by considering an occasion when the box is in uniform motion. Mr Smith tries to measure its velocity. He would like to select a point in space outside the box and to measure its velocity relative to that Joint. But he finds, to his disappointment, that he cannot even define a point in space in any helpful way, let alone measure a velocity relative to it.

One point in empty space is exactly like any other point. It can only be defined as its distance from a reference object and the only available reference object is the box. Mr Smith could choose a point defined as being at a particular distance from this. But then the point remains, by definition, always at the same distance from the box, whatever the speed of the box may be. If the box is considered to be moving, so is the point. If the box is at rest, so is the point.

Mr Smith cannot avoid the conclusion that he has no means of knowing whether his box is at rest or in uniform motion. It is, indeed, meaningless to ask the question. In the technical language of relativity there is no privileged frame of reference by which to define a uniform velocity. Mr Smith is perfectly free to ascribe any velocity that he likes to the box. The figure that he mentions cannot be disputed.

Relativists have another way of expressing this. Let us suppose that someone has constructed a four-dimensional model of the space-time in which the box is situated. One cannot, of course, make such a model out of wood or metal; but one can represent it by algebraic symbols. For our purpose such a representation is just as serviceable. Nothing would be gained here by writing the symbols down. It suffices that it can be done and that the space-time relations can be expressed mathematically as clearly and precisely as they would be if a material model could replace the algebra. How this is done is not relevant.

If successive positions of the box in uniform motion are represented by points in the model the points lie along a straight line. This may be vertical or sloping; it depends on how the model is tilted. One can, if one likes, adopt the convention that zero velocity is represented by points that lie vertically above each other. Points along a sloping line would then represent a finite velocity, which would be greater the more the line departed from the vertical.

If Mr Smith knew which was the right side up for his model he could then read the velocity of the box off from it by measuring the slope of the line. But he finds that there is no means of determining a right side up. He is free to tilt the model in any way he likes. He can so tilt it that the line representing successive points of the box is vertical and say that it shows the box to be at rest. But he can also tilt it so as to show the box to be moving at a uniform velocity. No inclination of the model could be claimed to represent a wrong velocity, which is another way of saying that there is no such thing as absolute velocity.

This is the basic tenet of special relativity, but not, of course, the whole of it. What has been said above does not show the part played in special relativity by the velocity of light. But this omission need not detain us. Our immediate concern is with general relativity.

Let us next suppose that there is a hook in the ceiling of the box on the outside and that a rope is attached to this hook. Einstein asked us to imagine that the rope is being pulled and pointed out that it is relevant to say by what agency. For descriptive purposes it is, however convenient to name the pulling agent, so I shall say that an angel is pulling the rope from time to time and with a constant force. As a consequence the box, with Mr Smith inside it, receives at intervals a constant acceleration.

Action and reaction being equal and opposite Mr Smith is subjected to a force in the opposite direction from that in which the angel is pulling. While the pull is upwards away from the ceiling the force observed Mr Smith is downwards towards the floor. Being inside the box Mr Smith cannot see the rope and does not know what causes the force on his box He reasons as follows:

'A moment ago there was no force, now there is one. So there is a physical change to the system in which I find myself. What is the cause of this change? I can think of two possible ones. The first is that this box is being accelerated in the direction towards the ceiling. The second is that some undetectable device is attached to me and is pulling me downwards towards the floor. Have I any means of discovering which of these two theories is the correct one?

'Can my model of space-time help? If the first theory is the true one, my box is being accelerated, successive positions when plotted in the model will not lie on a straight line. They will lie on a parabola. Let me do the plotting. Now let me tilt the model in such a way that it represents my box at rest. I find that it cannot be done. I can tilt the model so that a little bit of the line connecting the points is vertical. But then all the remainder of the line is sloping. Velocity can be made to disappear on the model, but not acceleration. The inference is that a body in uniform motion can rightly be regarded as at rest all the time; but a body in accelerated motion can be regarded, at most, as at rest for a moment time.'

It now occurs to Mr Smith to construct a different model of space-time. He calls it model B and calls the first one model A. In model A equal distances in space-time were represented by equal distances in the model. In model B it is not so. Distances are represented to functional scales, somewhat like log scales on graph paper. The selected scales are unusual and represent relations in non-Euclidean geometry. They are arranged in such a way that successive positions of the box lie along a curved line when the box is in uniform motion and along a straight line when the box is being accelerated at a uniform rate. As for model A there is no reason why model B should be tilted in one way rather than in another. Like the first model the second one correctly represents the fact that there is no absolute velocity.

The second model can be so tilted that the points lie along a vertical straight line. It then represents a condition when the box is at rest. Let us learn how Mr Smith reasons about this:

'When I constructed my second model I thought of it only as a convenient plotting device that would enable me to place all the points along a straight line. I regarded model A as the true scale model and model B as one constructed deliberately to be out of scale, to be a distorted representation of reality. But can I have been wrong in this? Is model B more than a mere plotting device? Can it be a true-to-scale model of the space-time in which my box is situated? Can space-time have the strange shape that is represented by the model? If so space-time itself has a physical property, namely curvature.

'This suggests a third way of accounting for the force that I am experiencing. The reason may be neither that the box is being accelerated upwards nor that I am being pulled downwards. That notion that I am being pulled is unsatisfactory anyhow so long as I cannot detect any devices attached to me by means of which the pull can be transmitted to my body. So the third explanation, which has only just occurred to me, deserves consideration. It is that the box is in a region where the space-time is non-Euclidean.

If this is the explanation the box is not being accelerated. It can be regarded as either at rest or in uniform motion. Model B correctly represents the situation. I can tilt this model so that the line is vertical if I like. So I shall give up the old notion that something is pulling me downwards. It ought to be discredited anyhow. I shall consider instead only the two possibilities that remain, namely that the box is being accelerated in a space-time correctly represented by model A or is at rest in a space-time correctly represented by model B.

'Can I make any observation here and now by means of which I can decide between these alternative possibilities?,

'No. The force exerted on my body must be just the same whether I am being accelerated in flat space or am at rest in curved space. The only difference that I can mention is that I attribute the force to the inert mass of my body when its cause is an acceleration and I attribute it to a different property (I shall call it the gravitational mass of my body) when the cause is the curvature of space. But if the only difference between two properties is the name that convention has given them there is no real difference. So it is reasonable to regard inert and gravitational mass as identical.'

By gaining this piece of insight Einstein took a great step forward towards the unification of science. What has been said here so far conforms to the traditional view of general relativity and differs only slightly from the conventional manner of its presentation. But it now becomes necessary to consider some of the false conclusions at which one can arrive if one thinks carelessly and superficially. If any of those who speak on relativity theory with authority have in fact arrived at any of these false conclusions, as I fear may have happened, it has become very necessary to do some rethinking. If I am wrong and no errors have been made it will nevertheless help towards clarity to appreciate what errors could be made.

Mr Smith might be tempted for a moment to think that it was meaningful to speak of a 'field of acceleration' in the sense in which one speaks of a gravitational field. (After all, Reichenbacher did so.) Mr Smith would suggest therewith that a field of acceleration occurs in flat space when a force is the result of an acceleration just as a gravitational field is present in curved space. But he would, I hope, soon realize that such a manner of speaking was misleading and imprecise. The word 'field' should be retained for the condition of a region and not for what happens in it. The acceleration of a body does not change anything in its surroundings. If the space is flat it remains flat when something in it is being accelerated.

Should Mr Smith persist however in speaking of a field of acceleration he might fall into the further error of thinking that the space around him really did change its curvature when a body was accelerated in that space. Suppose he were to climb to the ceiling of his box and hang there from the strap for a short while. If he let go his body would be accelerated relative to the box. It would be so whether the box was being accelerated in smooth space or was at rest in curved space. If it was the former Mr Smith would be at rest after he had let go while the box would continue to be accelerated. If it were the latter Mr Smith would be accelerated after he had let go, while the box would continue to be at rest. By falling Mr Smith does not smooth out the space that surrounds him. He just falls.

The error of thinking that the act of falling smooths out space-time can only arise if one believes that the choice between model A and model B depends on the motion of the observer. But it will have become clear that it does not. As I have said already, Mr Smith can find nothing inside his box that will enable him to know which model to choose. His own motion relative to the box does not tell him; nor do any other observations made there.

Having appreciated the fact that he cannot alter the curvature of space-time around him by falling any more than he can by dropping his watch Mr Smith will replace his previous assertion by the following:

'When I feel a force on my body and still have hold of the strap my successive positions lie along a curved line if plotted in model A and along a straight line if plotted in model B. After I have let go I do not feel any force and the shape of the lines in the model is the other way about. While I am falling my successive positions are represented by a straight line in model A and by a curved line in model B. I can represent this n conveniently in tabular form as follows:


Circumstance

Condition of space-time

Smooth

Curved

Correct model
Body subjected to a force when
Body not subjected to a force when
Uniform motion represented by
Accelerated motion represented by
Force attributed to

If A
Accelerated
At rest
Straight line
Curved line
Inert mass

If B
At rest
Accelerated
Curved line
Straight line
Gravitational mass

I must refrain in future from saying, or implying, that my actions change the curvature of space-time. As I do not even know from any observation that I can make from where I am, whether the correct model is A or B, I can certainly not claim to know that when I am falling I am changing from the one model to the other.'

There is the possibility of yet another error. Mr Smith may reach the erroneous conclusion that he possesses no means at all of deciding between the two models. He may, in other words, believe that observations made by an observer outside the box are as inconclusive as those made by himself.

If he makes this mistake he will feel justified in asserting that the two models, as any others, are equally valid in the same way as two different tilts of either model are equally valid. He will then enunciate the theory of full equivalence between a body at rest in curved space-time and a body in accelerated motion in smooth space-time. But Mr Smith is fortunate in having a friend outside the box from whom he can obtain some useful information.

The friend's name is Mr Jones. He is standing beside the angel. The two gentlemen establish telephonic communication. The following conversation takes place:

SMITH: 'What is happening where you are?'
JONES: 'The angel is pulling himself hand over hand along the rope towards your box.'
SMITH : I believe you are wrong. It seems to me that the angel must be pulling the box towards himself.'
JONES : 'Have it that way if you like. All that I can observe is that the rope is taut and is getting shorter. It is doing so at a rate of 9.80665 metres per second, per second. The explanation might equally well be that the angel is being accelerated towards the box or that the box is being accelerated towards the angel. I have no means here of knowing which it is.
The two statements are fully equivalent. To ask which it is is a meaningless question.'
SMITH: 'But I have means of knowing which is which. I am experiencing a force towards the floor of the box. As my body has inert mass I conclude that it is the box and not the angel that is being accelerated.'
JONES : 'After what you have told me that seems to be the more probable explanation. My observation alone left the question open, but our joint observations make it reasonable to assume that the box is being accelerated and not the angel.'
SMITH: 'Yes. And my observation alone left the possibility open that there was no pull at all on the rope but that, instead, the box was in a non-Euclidean region of space-time. Our combined observations have provided the answer to that question too.'

At this moment it occurs to the two observers that Mr Jones would have known whether the angel was being accelerated or not if the angel had possessed inert mass. They agree that acceleration of a body P relative to Q is not equivalent to acceleration of Q relative to P. If the bodies are in smooth space-time the force on each is the product of its acceleration and its inert mass. If the acceleration is zero its product with inert mass is also zero. Conversely, if the force is zero and the space-time is smooth the acceleration is also zero.

After a while the angel gets tired of intermittently pulling at the rope. He changes his position and places himself underneath the box. Mr Jones accompanies him. Presently Mr Smith calls Mr Jones on their telephone.

SMITH: 'Jones. The angel is pulling at that rope again.'
JONES : 'No he is not. The rope is curled up neatly and lies on top of the box. The angel is underneath the box. I am here with him.'
SMITH: 'What has the angel been doing?'
JONES : 'He has placed a big sphere underneath the box. Its circumference is 40,000 kilometres.'
SMITH: Is that all?'
JONES: 'Yes. All I can see.'
SMITH : 'As the angel is no longer pulling at the rope I must assume that the box is no longer being accelerated. And yet I feel that same force as I did while the acceleration lasted. So I have to conclude either that some undetected device is attached to my body and pulling downwards or that the box is in a non-Euclidean region of space-time. I deprecate the mystical hypothesis of undetectable devices and so I have to assume the latter. Something must therefore have changed the physical nature of space-time around here in such a way as to cause curvature. The only physical change that you can report is the appearance of the big sphere. So I conclude that this is influencing the geometry of space-time in its vicinity.'

Mr Smith and Mr Jones confer further together and reach agreement on several points. They are satisfied that there is no difference between inert and gravitational mass; that there is nevertheless a difference between acceleration and uniform motion; that relative acceleration has real meaning; that the process that causes a stone to be subjected to a force during its acceleration is not the same as the process that causes it to be subjected to a force while it is at rest on a shelf; that an observer may be conceived to be so situated that he cannot discover which of the two processes is operating; that an observer may also be so situated that he can discover just this; that, for instance, Smith inside the box can only observe the effects of the processes, which are identical for both, while Jones outside the box can observe the sources of the processes, which are different for both; that it is the big sphere underneath the box that causes the geometry of space-time to become non-Euclidean.

Having thus clarified the subject they are able to formulate the big question that had previously been obscured by sundry misconceptions:

By what process is the distortion of space-time effected?

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