TOWARDS A UNIFIED COSMOLOGY

by     Reginald O. Kapp

PART V - APPENDICES

Appendix H - Space and Mass


H.1: The Physical Properties of Space
When, early in this century, relativity theory made its tremendous impact on scientific thinking, one of the most revolutionary and profound conclusions to which it led was that the relation between space and matter had to be re-defined. As I have emphasized in Chapter 21 and elsewhere, we had until then thought of the relation as that between container and content. The container, space, was assumed to have no property other than extension and all distinguishing features that differentiate one thing from another were thought of as belonging to the content.

The content was thought of as comprising various constituents. There were, of course, ponderable matter and radiation. But not only these. Any given quantity of ponderable matter has an environment, and in this distinguishing features are provided by fields of force. These are yet another component of the material universe and were regarded in pre-relativity days as a part of the content that is accommodated in the container, space. They must occupy our attention for a short while.

Fields of force are measured and described in terms of their intensity, which is more precisely called their potential gradient. There are three kinds of field, electrostatic, magnetic and gravitational; so the environment of a given quantity of ponderable matter may contain three kinds of potential gradient. Each of these may vary both in magnitude and direction.

The potential gradients have a physical effect on ponderable matter, namely that of accelerating it; so they are physical realities. It follows that the environment of a given quantity of ponderable matter has physical properties.

In pre-relativity days these properties could not be attributed to space, as this was considered to have no property other than extension. So they were attributed to something else in the environment, something called luminiferous ether. This was regarded as a kind of matter and as yet another of the things that were accommodated in the container called space.

Like space, the hypothetical ether was endowed with the property of extension but was believed also to possess further properties, among them elasticity. By virtue of these it was supposed to be subjected to a strain in places where there was a potential gradient, the amount of strain being proportional to the gradient.

This view has had to be abandoned. One of the pieces of insight gained from relativity theory is that it means nothing to speak of a con- tainer for the whole material universe. The notion that space serves this one and only function is, therefore, not held today. We know now that such space would be but a meaningless abstraction.

Nevertheless, a name is needed for the physical reality that surrounds any given quantity of ponderable matter. 'Luminiferous ether' might have been retained for this were it not that its verbal currency had been depreciated by sundry misapprehensions about it. Hypothesis had, for instance, endowed the ether with a variety of properties, of which elasticity was but one; density, mass and others had also been spoken of. But no such properties could be attributed to places where there was no ponderable matter. The only observed, or even inferred, physical features of the environment were various potential gradients, and so the many properties that had been attributed to the ether proved both inappropriate and too numerous to make this term suitable. If the word 'space' suggested some- thing too abstract, the words 'luminiferous ether' suggested something too concrete.

Perhaps the non-committal word 'environment' might have been a good choice. It could have been justified scientifically; for it is strictly accurate to say that a quantity of ponderable matter is surrounded by an environment that has physical properties and one has to be very cautious about saying anything more than this. But to use 'environment' in this way would have been to raise the word to the status of a technical term and to use it, moreover, in an unfamiliar context. It would probably never have succeeded in ousting the established word 'space'. So I think that Einstein showed a sound intuition when he retained this word while changing its meaning from that of a featureless container to that of a synonym for featured environment. But when the word 'space' is used in relativity theory with its new and precise meaning, it does not at the same time represent the vague concept that it does in everyday speech. It has become a technical term.

Another piece of insight gained from relativity theory was that the properties previously attributed to the luminiferous ether could not be attributed to the environment of ponderable matter. They were all replaced by one single property described as 'curvature'. It was from the identity of inert and gravitational mass that Einstein arrived at the conclusion that a field of force, at least when the force is gravitational, is a region for which the geometry of space is non-Euclidean, and he used the expression 'curvature' as a means of describing the departure from Euclidean geometry of the space in which there is a potential gradient. This, too, became a technical term with a unique and precise meaning in relativity theory.

The conclusion startled the scientific world. Until then we had regarded curvature as a purely geometrical property. That it could be regarded as a physical one caused scientists to revise their notions about the nature of matter. How, it came to be asked, can something as apparently abstract as a curvature of space interact with something apparently as concrete as a material particle? The question is as puzzling today as ever it was.

The best answer will eventually come, I am inclined to think, from a revision of our notions about what is abstract and what is concrete. In making a distinction between the things to which we think the two terms appropriate we are enslaved by the organs of sense perception with which we, as human beings, happen to have been endowed. We call those things concrete that we can perceive directly with the help of these organs and those things abstract that cannot be so perceived. But this is a surmise and I do not propose to pursue this line of thought any further here.

Relativists have succeeded in showing good reason why gravitational potential gradients should be identified with a condition of space appropriately called its curvature. But they have not yet succeeded in showing the same for electrostatic and magnetic potential gradients, though they still hope to do so. The effort to do this is called the search for a unified field theory, and I have briefly alluded to it in Chapter 21. Anything that I, or anyone else, can say today about the relation between space and matter may have to be modified if and when the search has succeeded. For this reason, if for no other, what is said in this appendix must inevitably not only leave some insistent questions unanswered, but also be very tentative.

Should it ever be proved that electrostatic and magnetic potential gradients are regions of curved space, we must expect the kind of curvature to differ basically from the kind that is identified with a gravitational potential gradient, for there is no interaction between gravitational fields and the other two kinds. A change of the charge on a body having inert mass does not change the behaviour of the body in a gravitational field. A change in the potential gradient of a gravitational field does not affect the forces between electric charges placed in it. Gravitational masses that do not carry electric charges and are not magnetically polarized fall with the same acceleration through an electrostatic field (provided there is a gravitational one to fall in) irrespective of its intensity. In other words, electric charges behave in the same way in gravitationally curved as in flat space.

From such observations it has to be concluded that, if electro-magnetic potential gradients are, like gravitational ones, curved regions of space, the geometry by which the curvature can be defined can hardly be of the same Riemannian kind as represents gravitational potential gradients. This independence from each other of the different kinds of field takes some accounting for and even leaves room for doubt whether the electro-magnetic field is, as seems so reasonable to suppose, a region where the geometry is non-Euclidean. But I have nothing to contribute to the search for an explanation and do not propose here to discuss the relation between space and charge or that between space and magnetism, but only the relation between space and mass. I have no choice but to discuss this relation as though the only condition of space by which one region is distinguished from another is curvature and the only known kind of curvature is that identified with the gravitational field. The incompleteness of such treatment is made obvious by the known facts about electricity and magnetism. But all that one can do in the present state of ignorance is to see along what path the incomplete treatment takes one and to hope that better knowledge of the relation between space and charge will not necessitate too great a change of direction from that in which the chosen path leads.

H.2: Interaction between Space and Mass
I want here to define the limits to our present knowledge of the relation between space and mass. Relativity theory has contributed a great deal to that knowledge and I am hoping that Symmetrical Impermanence can contribute a little more. But there is still much about which we are ignorant and questions to which the search for answers is likely to prove rewarding come under two headings: the action of space on mass and the action of mass on space.

The Action of Space on Mass: Relativity theory tells us something about the motion of a particle to which no force is applied. The motion depends, according to the theory, only on the curvature of the space in which the particle finds itself. If the space is flat, the particle moves with a constant velocity, which may of course be zero velocity. If the space is curved, the particle moves with a non-uniform velocity; it experiences an acceleration or a deceleration.

If a force is applied to the particle, its movement is no longer wholly determined by the geometry of space. A billiard cue can accelerate a billiard ball in flat space and a shelf can, by exercising a force on the stone resting on it, prevent the stone from following the curvature of the space-time in which it finds itself.

I mention these well-known facts only to bring out what is and what is not known about the action of space on mass. Relativity theory defines the property of space by virtue of which it is able to act on mass; this property is technically called curvature. But relativity theory does not tell us anything about the property of mass by virtue of which an unrestrained particle follows the curvature. The property is given the name inertia. But to name a thing is not to explain it or to give any sort of information about it. Relativity has some important things to say about space but none about mass. Let me illustrate our present ignorance about the relation between these two concepts with the help of an analogy.

When one sees a trarncar turn a corner, one may give the perfectly true explanation that there are rails for it to travel on. But it is an insufficient explanation. It says something about the street, but nothing about the tram. To make the explanation complete one has to add that the tramcar is provided with flanged wheels, so designed that they fit the rails.

Similarly, when one sees a stone fall, one may give the explanation that the space in which the stone happens to be is curved and thereby causes a curved track in space-time. One has then found the equivalent of the rails that take the tram round a corner. The analogy is admittedly not perfect, for a force is exerted between the tram rails and the flanges on the wheels, whereas the stone follows the curvature of space without any force being exerted on it at all. But the analogy serves, nevertheless, to show in what way the relativistic explanation remains incomplete. It fails to include the equivalent of the trarn wheels. What, one is led to ask on receiving the relativistic explanation, is the feature of a particle that causes it to 'engage' with space, as it were, so that its track in space-time follows the curves? In the example of the trarncar, steel wheels run on steel rails. When an unrestrained particle moves in space, something is said by relativists to run on curvature. What is it?

I shall return to the question later. For the moment I want to formulate some others.

The Action of Mass on Space: For the sake of convenience I shall again repeat a few well-known facts. According to Newtonian mechanics a massive body causes other bodies in its vicinity to be accelerated. To the question why this happens the answer is that a field of force surrounds the body. But to the question: what sort of a thing is this field? Newtonian mechanics has no answer; it can at most provide a name. Newton did not attempt to explain gravitation; he was content to postulate it.

It is here that relativity theory steps in. It does say what sort of a thing a gravitational field is, namely a region of curved space. The answer is justified both because it is methodologically sound and because it has considerable explanatory power. But it leads to the further question: why should a massive body cause the space around it to be curved? Again, relativity says something about space, but nothing about mass. It is content to postulate the effect of mass on space without explaining it.

It is here that Symmetrical Impermanence, in turn, steps in. The answer that it gives has been developed in detail in Part Four. It is that the extinction of an elementary component of the material universe is accompanied by the extinction of some space and that this latter gives rise to a wave of contracting space, manifest as the kind of curvature that acts on mass. This answer leads, in turn, to another question: why should the extinction of an elementary component be accompanied by the extinction of some space? Symmetrical Impermanence does not attempt to explain this coupling of space with mass; it is content to postulate it.

There is admittedly some justification for doing so. If matter were to become extinct while space did not do so, the average density of matter in the universe would decrease without limit; as it also would if space were to originate while matter did not. It was acceptance of this idea that space is originating (the expanding universe theory) coupled with the postulate that the mass density of the universe remains constant that led Hoyle, Bondi and Gold to postulate the hypothesis that became known as that of 'continuous creation'. Similarly, the average density in the universe would increase if space were to become extinct while matter did not, or if matter were to originate while space did not.

But if the assumption of this kind of coupling of space and mass can be justified, it has not yet been explained. Like relativity theory, Symmetrical Impermanence accepts a two-way interaction between space and mass without doing much to explain it. Yet there can be hardly any doubt that an explanation, if found, would add much to our understanding of the nature of matter.

H.3: Some Questions about the Nature of a Particle
The ignorance to which I have just been drawing attention is all about the nature of a particle. To illustrate how great this is I give below some sample questions. At the present stage of science satisfactory answers cannot be given to any of them. There is no more than a somewhat frustrated groping. Some of the questions have just been formulated above, some are well known, some have a particular bearing on the theme of this book. Each of these questions needs to be pondered over. None can be dismissed lightly.

(1) What in the nature of a particle causes it, in the absence of restraint, to follow a track that is determined by the curvature of space?
(2) What in its nature causes the particle to depart from this track when it is subjected to an impact?
(3) Why is the outline of an elementary particle indeterminate in such a way that the particle has some of the properties of a wave?
(4) What causes a particle to cohere with others in an atomic nucleus with a force sufficient to overcome the forces of repulsion between the positive charges in the nucleus?
(5) Why should a particle that collides with a nucleus sometimes disrupt the nucleus and sometimes be captured by it?
(6) Why does the ratio of mass to charge in a stable nucleus increase with increasing atomic number?
(7) How can, as is postulated by Symmetrical Impermanence, some of the mass of a substance become extinct without either producing radio-activity or leaving a residue of atoms of lower atomic number?
(8) Why are origins of space and of mass coupled and, similarly, extinctions of space and of mass coupled?

During the many years while I have been exploring the implications of Symmetrical Impermanence I have necessarily given much thought to all the questions in the above list. I could not, in particular, afford to ignore questions (1), (2), (7) and (8). If the facts that lead to these questions are not true, most of what appears in this book is invalid; so I should have liked to discuss only those four here and say nothing about the others in the list. But I have found that such isolation is not possible. The questions are so linked that the answer to one seems to lead to answers to the others. Hence all or none must be discussed.

Therewith I find myself on the horns of a dilemma. It has not been easy to decide whether I ought to publish here and now my tentative conclusions about the nature of a particle or not.

The main argument against doing so is a strong one. It is that I have not given the subject the same number of years of careful thought that have gone to the conclusions presented in Parts One to Four of this book. There I was able to introduce statements with such words as 'it follows that'. The statements were, so far as I could make them, inferences. If, after they have been scrutinized by others, they prove to be wrong, it can only be because of my faulty reasoning. But if I now discuss the nature of a particle I shall be obliged to introduce some statements with the much deprecated word 'perhaps'. If these statements are found to be wrong, it may not be only because of my faulty reasoning; it may be because of a less excusable indulgence in speculation. In other words, I cannot promise entirely to avoid ad hoc hypotheses.

Nor can I promise anything that deserves the title of a theory. The conclusions at which I have so far arrived take me no further than to the beginning of a path that seems to lead in a promising direction. It could be urged with considerable justification that I ought not to invite others to tread this path until I have explored it further myself.

Be it added that the nature of a particle is a very big subject. It is much too far-reaching to be adequately discussed in the final appendix to a book concerned with quite different problems. There are indeed good arguments for postponing discussion of this subject to a later occasion.

But there is the other horn of the dilemma. The relevance of questions (1), (2), (7) and (8) to the theme of this book has already been mentioned. Question (7) has been discussed in Appendix G and question (8) is an immediate offspring of the new theory of gravitation. Thus both these questions are new; and new questions have a peculiar insistence. While one is giving one's attention to the implications of Symmetrical Impermanence, these two questions tend to assume a significance greater than any of the others. It may well be an exaggerated significance. With a proper sense of proportion one would, I think, give pride of place to other questions in the list. But in the immediate context these two ought not to be ignored. Some may, I fear, regard it as a waste of time and effort to give any attention at all to the conclusions reached in this book unless someone can first provide them with answers to questions (7) and (8). These two questions may, in other words, provide a pretext for ignoring my main thesis, whereas they ought really to act as a challenge and stimulus to explore the subject further.

This consideration has eventually, and after several changes of mind on my part, outweighed my disinclination to publish conclusions that are still tentative. So the remainder of this appendix will be concerned with a highly condensed presentation of a vast subject. I shall try to show that there is quite a good prospect of taking the unification of physics a stage further by giving concentrated attention to the eight questions listed above.

H4: Bound and Free Curvature
The kind of electrostatic field that is measured by the force between charges is bound to the charge at its centre. An electron carries such a bound field around with it. So long as a conductor does not move in space and its charge does not change, the electric potential gradient at a given place does not change with time. If the electrostatic potential gradient is a region of curved space, it is one where the curvature can be called 'bound'. This curvature is represented to the imagination as a sort of halo surrounding the electron or proton and it will be convenient for the present purpose to use this term. I do not do so in a derisory sense.

An electrostatic field can also be free of a particular charge. It then travels, in partnership with a magnetic field, in the form of radiation. This happens when the two fields break off from the antenna of a wireless transmitter. As the radiation passes a given place, the electrostatic potential gradient there changes with time. One can call it 'free'.

The question now arises whether gravitational fields are analogous to electrostatic ones in that they occur in both the bound and the free condition. If so, one can speak of bound and free curvature.

According to the traditional relativistic theory of gravitation, it is not so. Only bound curvature is recognized. A neutron is supposed to carry a very faint halo of gravitational curvature around with it, analogous to the strong electrostatic field carried as a halo by the proton. This latter is supposed to have two halos, the faint gravitational one in addition to the strong electrostatic one. The gravitational curvature is assumed never to occur as a free travelling wave, but always as bound to the mass at its source.

According to the new theory of gravitation that has been presented in Part Four this is denied. The neutron has no gravitational halo and the proton has only the electrostatic one. For gravitation does not come into existence until the object at its source becomes extinct. The field in which a stone falls, it will be remembered, consists of a succession of pulses of contracting space, which are the consequence of extinctions and travel outwards from the source of the field. Thus the gravitational field is regarded in the new theory as analogous to free electromagnetism, as analogous to the light that emanates from a lamp. Each photon that contributes to the illumination is detached from the lamp and travels freely.

Thus the traditional theory recognizes only bound gravitational curvature and it has been made to appear in Part Four that gravitational curvature is always free. But is this correct? If one accepts the new theory of gravitation, must one assume that the kind of gravitational field in which a stone falls is the only one ? Or is it consistent with the new theory to assume that bound gravitational curvature is a physical reality and occurs in addition to free curvature? Are there gravitational fields that are not composed of waves and in which the potential gradient does not change with time?

I have already said that I shall permit some 'perhapses' to enter this appendix, even though I have hitherto deprecated them and have done my best to keep them out of the main parts of this book. Let the first 'perhaps' be one that occurred to many of us in the early days of relativity: perhaps an elementary particle is, apart from its charge, a region of gravitationally curved space and nothing else.

Eddington made this point a fundamental one. 'Mass is curvature', he said somewhere. At the time when he was writing there was a much used catch-phrase: 'Man is a kink in space-time'. Preoccupation with this hypothesis is less insistent today, but not because we are any nearer either to accepting or discarding it. It is only because we are today more concerned with the way particle acts on particle than with the way space acts on particle. The great attention paid to nuclear physics has diverted attention from the fundamentals of relativity. But I do not think that what is at most a slight lack of topicality makes the question whether mass is curvature or something else any less rewarding now than it has been in the past.

To avoid the complication of electricity and magnetism I shall begin by considering only a neutron. What is it made of? Must we postulate a substance that is different in nature from the technical concept, space? Shall we find ourselves obliged to speak of 'particle stuff'? Or shall we say, with Eddington, that the neutron consists of curvature and nothing else? When the question is put with this disconcerting candour one is inclined to dislike every answer that can be suggested. But I think that one will have the greatest dislike for the suggestion that there is something deserving of such a title as 'particle stuff'. One will be inclined to accept the Eddingtonian answer, if only as the lesser evil. The theory that the neutron consists of curvature seems better to meet the Principle of Minimum Assumption and to offer a better prospect of further unification of physics.

If this is accepted, the volume occupied by a neutron is a region of bound gravitational curvature. Within this region the potential gradient does not change with time in the way it does when the curvature takes the form of a travelling wave of gravitation.

To say this is probably to say much the same as was almost taken for granted by relativists forty years ago. But there is a significant difference. Traditional relativity had to be a little vague about the nature of the curvature. It has to be remembered that it postulates what I have called a faint halo around the neutron, a region of gravitational curvature with unlimited extent and flattening with distance in accordance with the inverse square law. So far as I know, there was no discussion as to whether the mass of an elementary particle was this faint halo and nothing else, or whether the particle was a composite affair, consisting of the halo in combination with an inner core of more intense curvature. In those days the time was not ripe for discussion of this question.

To be consistent with the new theory of gravitation one has, however, to face it. As the new theory denies that a gravitational field surrounds every neutron, one must regard the neutron as being a simple, and not a composite structure, as consisting of intense curvature in the volume occupied by it and as free from the feeble curvature previously postulated in its environment.

Therewith we arrive at a new way of distinguishing between space and mass: Electromagnetism apart, space consists wholly of free curvature and mass consists wholly of bound curvature. For the word 'curvature' one can, without change of meaning, substitute potential gradient. A way of understanding this will be presented later in Section H.7.

The implications of this conclusion are manifold and I have no doubt whatever that they deserve prolonged study, even if such study eventually leads to rejection of the hypothesis that mass is curvature. But this is not the occasion for a detailed inquiry. I must be content to mention one implication very briefly here and some others in the following sections.

I have said in Section H.2: ‘In the example of the trarncar, steel wheels run on steel rails. When an unrestrained particle moves in space, something is said by relativists to run on curvature. What is it?' If mass is curvature, the answer is found. Only curvature can run on curvature. It seems as reasonable a conclusion as one may hope for so long as one accepts general relativity.

H.5: Space and Anti-Space
If the curvature of which a neutron consists were unfolded, how much space would it occupy?

Space and mass are so coupled, it has been mentioned repeatedly here, that when some space becomes extinct so does some mass; and when some space originates some mass originates also. There is thus an equivalence between a given quantity of space and a given quantity of mass. It should be possible to arrive at the number by which these quantities are related.

The coupling of space and mass has been justified here, it will be remembered, by the argument that the density of matter in the universe would tend either to infinity or to zero if there were no such coupling. To maintain constant mass density the quantity of space that becomes extinct whenever a neutron does so must be the quantity that is, on the average, occupied by the mass of a neutron. Estimates of this quantity vary widely and are difficult to arrive at with any degree of certainty. A typical figure for the density of the universe seems to be the mass of one nucleon (proton or neutron) per cubic centimetre. Be it then said as a rough approximation that when the curvature constituting a neutron becomes extinct, the volume of one cubic centimetre does so too.

What is the connection between the two extinctions? Does the one trigger off the other, or is there a conversion from one thing into something else?

The triggering-off hypothesis would require for its elaboration the assumption of some rather complicated mechanisms; it would not meet the Principle of Minimum Assumption. So I prefer the conversion theory. But what is converted into what?

If one assumes that a quantity of curved space constituting the neutron is converted into a quantity of flat space, that there is an 'unfolding' of the space of which the neutron consists, then a cubic centimetre of space would not become extinct when a nucleon did; there would be an addition to the space that existed already. The conversion that we have to look for is one into something that cancels existing space, not into something that adds to it.

This consideration leads me to venture a further 'perhaps'. It is that there are two kinds of space, suitably named respectively 'space' and 'anti-space'. Such a distinction would complete the duality that seems to be observed for all the basic concepts in physics: positive and negative charge, N and S magnetism, the electron and the positron, the proton and the anti-proton, matter and anti-matter, positive and negative energy. Perhaps the duality of space and anti-space could be shown to comprise in itself all the others.

It is a bold assumption but it has a good deal of explanatory power. An inference from it is that an elementary particle is not a curvature of space, but a curvature of anti-space. A quantity of this, say one cubic centimetre, is curved or folded into the volume of a neutron or a proton. What I have called the extinction of an elementary component of the material universe is then the unfolding of this quantity of curved anti-space. The flattening out is equivalent to the extinction of an equal quantity of space and is manifest as a wave of contraction.

The hypothesis is not complete unless it is applied to origins as well as to extinctions. To be consistent one must regard the origin of space, which is manifest in the expansion of the universe, as the consequence of the unfolding of a corresponding quantity of curved space. What unfolds must be something that could appropriately be called an anti-particle and differs from a particle in consisting of curved space instead of consisting of curved anti-space. The unfolding of bound curvature of space leads to the effect that I have called anti-gravitation in Part Four. The relation'. are presented in the following table:

 

CONSTITUENT

 

Curved Space

Curved Anti-Space

Effect when bound
Effect when free

Anti-particle
Anti-gravitation

Particle
Gravitation

I should shrink before the novelty of this conclusion if there were not considerable justification for it. But it seems to have the virtue of much unifying and explanatory power. It brings gravitation into a rational relation to space and it will appear from Section H.8 that it also helps to establish a rational relation between particles and space. The justification for the above table is, in fact, largely to be found in Section H.8

. Considered quantitatively the assertion that a particle unfolds into a cubic centimetre or so of space (or anti-space) is rather startling. We all know that a curved line can be measured in two ways: round the curve and across the straight line that joins the ends of the curve; and we know that the first measurement gives a larger value than the second. We can accept the statement that curved space is analogous to a curved line in that its volume can be measured in two ways; and we can believe that the one measurement gives a greater value than the other. Relativity theory has accustomed us to this notion. But for the gravitational fields around the earth or the sun the difference between the two volumes is small. What many may find difficult to accept, I fear, is that space can be so intensely curved back on itself that the difference between the two volumes is that between one cubic centimetre and the volume that we assign to a neutron. Yet this is what I am suggesting.

To those who find this disquieting I should point out that if we are committed to the notion that a particle is curvature and nothing else we are also committed to the notion that the curvature is very intense. We ought not to expect a gentle curvature of space to manifest those properties (hardness, capacity for hitting things, and so on) that we observe in a particle.

The question arises whether anti-particles can be observed and, if so, what their properties are. It is worth asking, but it is arguable that one should not expect to observe an anti-particle. I should prefer to regard particles and anti-particles as occurring normally as couplets. Their properties would exactly cancel each other and they would have no effect of any kind on their environment. To all intents and purposes they do not exist so long as they remain as couplets. So it is meaningless to ask if there are few or many, or where they are. I am inclined to think that it is meaningless even to ask whether they exist, so long as their existence would have no effect on anything.

Occasionally, at random, and without cause, one of the two components of the couplet becomes free, the one that consists of curved space.

Becoming free means here collapsing into a wave of expanding space, into a pulse of anti-gravitation. The other component of the couplet, the one that consists of curved anti-space is left behind and now becomes manifest as a particle. One might say that it originates at this moment; for nothing was previously observable. But it might be better to say that the particle is uncovered when its companion, the anti-particle collapses. Uncovering renders it effective and observable.

Though, as I have just said, it is meaningless to say that the intact and unobservable couplet exists, it is meaningful, after the anti-particle has collapsed and the particle been revealed, to say that the couplet has existed. The point is a metaphysical one, but worth appreciating.

Before proceeding further I should like to suggest another possible way of interpreting as conversions what seem like origins and extinctions of matter.

It is a commonplace in general relativity that the value of π is not the same in a gravitational field as it is in flat space, the reason being that the field is a region of non-Euclidean geometry. For the field around the earth the departure from the value of π as calculated for Euclidean space is very small. It is only just observable for the sun's gravitational field. But these are fields where the curvature is free and the curved anti-space has gone a long way towards flattening itself out. The geometry in these regions is nearly Euclidean.

If a cubic centimetre of Euclidean space is crowded into the tiny volume of a neutron, the departure from Euclidean geometry within the neutron is very considerable. If one expresses its volume as V = (4/3) xr3, one must assign to x a value that differs from π by a very large factor indeed. It would, I venture to suggest, be rewarding to apply the mathematics of relativity to a space that was so extremely non-Euclidean as I am claiming for the inside of an elementary particle. Our understanding of the nature of a particle would be greatly enhanced thereby.

There is one suggestion in particular that I should like to make to anyone prepared to tackle this job. It looks on the surface as though x in the above expression for the volume of a sphere in highly curved space must necessarily be very many times greater than π. But is a geometry logically possible and consistent with the facts for which x would be very many times smaller than π? If there is it would presumably represent a curvature that differed from the assumed one by a plus or minus sign, This would lead to a distinction between curvature and anti-curvature, which might usefully replace the distinction that I have suggested above between space and anti-space. Instead of saying that a neutron was a curvature of anti-space, it might be methodologically preferable to say that it was an anti-curvature of space. The hypothetical couplet that I have postulated as the unobservable parent of a simultaneous wave of anti-gravitation and a new particle would then be a couplet consisting of a combination of curvature and anti-curvature. As these would cancel each other's effects, the couplet would be literally indistinguishable from flat space.

One would then have even more reason to deny the existence of the couplet before its separation into two manifest components. An origin would be really the origin from nothing of a wave of free curved space which would leave behind it, as its counterpart, the region of bound anti- curvature that we call a particle.

In good time this would collapse and disperse as a wave of free anti-curvature, called a pulse of gravitation, and it would be as though nothing had been. After the passage of the two events only flat space would be left.

If this interpretation is tenable, there are, apart from electromagnetism, two kinds of basic process in the physical world. The first of these is the separation out of flat space of two things: one is a wave of free curvature and the other a minute region of bound anti-curvature. These are distinguishable only in that the one is free and the other bound. The second process is the subsequent collapse of the bound anti-curvature. In collapsing it appears as a wave of free anti-curvature, which is equal and opposite to the previous wave of curvature. It differs from it, however, in time and space. If the two waves coincided in time and space, there would be no effect. Physical events occur only, according to this metaphysical interpretation of the relation between space and mass, because two processes that are equal and opposite are separated by a space-time interval. The very existence of a physical universe depends on this random, indeterminate, and uncaused interval of time between the origin of a wave of curvature and the later cancelling wave of anti-curvature.

These are very bold speculations and I should be surprised if they will survive without drastic amendment. Such ideas ought not as a rule to be presented to others to work through before their author has done much work on them. But there are exceptional occasions when one is justified in suggesting a line of investigation to others rather than in keeping it to oneself, and I regard the present occasion as one of these.

For one thing it is important to show that question (8) is one to which it is worthwhile to seek an answer and my way of showing this is to produce possible ones. I should rather make suggestions that may have to be discarded than leave people with the impression that the question is too difficult even to tackle.

For another thing, I have wanted to illustrate the rather important point that progress in basic physics cannot be expected without careful attention to metaphysics. The scorn that is sometimes cast on this discipline is, I feel sure, usually misplaced. If the hypotheses that I have put forward very tentatively about curvature and anti-curvature prove invalid, I shall regard them as having served a purpose if they direct attention to the crying need for bold and, I must add with emphasis, metaphysical thinking.

H.6: The Notion of a Graphical Symbol
Much in the last section recalls what has been said in Chapter 23 about the distinction between deductive and representational understanding. A lot of space in a small volume may be understood deductively, but not representationally. The basic concepts of physics are, it will be remembered, by their very independence of any particular observer, beyond our powers of representational understanding. Any effort to understand them in this way is therefore misplaced.

This conclusion raises the question of models suitable for representing basic physical concepts. Lord Kelvin clung firmly to the view that such models are always possible. He even went so far as to deny physical reality to a concept of which he could not make a model. But in Lord Kelvin's day the need in physics to 'eliminate the observer', the importance of this kind of objectivity, was not yet appreciated. It has brought about a revolution in methodology. It could still seem quite obvious to contemporaries of Lord Kelvin that a statement about basic concepts in physics could only be true if one could represent it by the kind of model that would have a meaning for the human imagination. But it has now become clear that the opposite holds: the statement cannot be both true and basic if it can be so represented; for then the observer has not been wholly eliminated. What can be represented in the human imagination, it will be recalled from Chapter 23, depends on the particular organs of sense perception with which homo sapiens happens to be endowed. But a basic statement in physics must not depend for its validity on this fortuitous biological circumstance; it must be distinct from any of those things to which the human imagination has access.

This somewhat disquieting piece of methodology is relevant here because it applies forcibly to the relation between space and mass. Curved space cannot be represented by a model that the human imagination can grasp and neither can the process by which the curvature of space influences the motion of a particle in it. One may hope some day to understand the relation between space and mass deductively, but one can never hope to understand it representationally.

Yet it is only human to try to make models of all one's concepts and some have been made for elementary particles. They are by no means to be deprecated; for each has its use. Even if they suggest some things that cannot be true, they also suggest some things that are. When, for instance, one is concerned with collisions between elementary particles, one represents each particle by a model that is a small hard sphere. The imagination pictures this as round, elastic, slightly deformable (like a tennis ball when it bounces), and as unbreakably strong, although the word used for this latter property is the rather evasive one 'indivisible'.

Such a model is helpful up to a point and represents at least a part of the truth. But there are occasions when it has to be abandoned in favour of a very different one, a bundle of waves, perhaps. The various models contradict each other and it is right and proper that they should. But this means that the word 'model' is not well chosen. I should like to see it replaced by a different word. The limitations of efforts at representation would thereby be recognized.

May I recommend the term 'graphical symbol'. It is exactly what we often mean when we now say 'model'. We may represent an electron by a circle on the paper and by a sphere in the imagination, but we intend no more than a symbolic way of representing some features of the electron. When we think of a blurred outline for the electron we intend no more than a symbolic way of representing its wave properties. When we speak of curved space and represent it by a curve on a sheet of paper we intend no more than a symbol for the potential gradient that occurs in the environment of a given quantity of ponderable matter. When we speak of transverse electromagnetic waves we intend no more than a symbol for a periodic change with time that occurs at right-angles to the direction of propagation of the waves. Sundry statements in quantum mechanics come into the same category.

In such instances we are only too often misled by the symbol into thinking that it is a picture of reality, differing from the thing symbolized in nothing except scale. The term 'model' encourages us in our error. The term 'graphical symbol' would be a salutary corrective.

The recommendation that 'graphical symbol' should become a recognized technical term in the methodology of physics deserves a longer essay than there is space for here. My reason for introducing it is that I want in a moment to suggest a new way of regarding an elementary particle and I want at the same time to avoid the impression that this, or any other way, can convey the truth, the whole truth and nothing but the truth. I want to go further and insist that this way, like all others, sometimes misleads. I want to propose little more than a new graphical symbol, but one that will convey features of an elementary particle that are not conveyed by the symbols conventionally used.

H.7: Point Symbols A graphical symbol is needed to represent, not necessarily pictorially but symbolically, the notion that a particle is a region of curvature (or anti-curvature), that its boundary is approximately but not quite precisely defined, that it is accelerated in curved space and so on. The symbol shown in Fig.7A would serve. It needs a name and the shape suggests 'point symbol'. It is thereby distinguished from the more familiar circle, which could be called a 'sphere symbol'.

A. In flat space and moving with uniform velocity; no force, no acceleration
B. In flat space and accelerated by an applied force; force and acceleration
C. In curved space and free from restraint; no force, acceleration.

D In curved space and prevented by a restraint from being accelerated; force and no acceleration
E. Elastic collision between neutrons; mutual forces and accelerations.

Fig. 7. Point symbols of a neutron

In interpreting the symbol the following conventions apply:
Any horizontal line represents flat space and any departure from the horizontal represents a curvature of space. (In spite of the tentative suggestions in section H.5. I shall speak here of curvature and of space. That will not preclude the substitution of the word 'anti-curvature' or 'anti-space' later, if either is found to be preferable.)
The distance 'a' represents the volume of the particle. It is analogous to the distance in flat space between two points that adjoin the particle on opposite sides. As the point is shown to rise gradually from the horizontal, the diameter is not represented with precision; the graph therewith represents symbolically the known uncertainty about the diameter of an elementary particle. If the bend where the graph rises from the horizontal were analyzed in a Fourier series it would give a collection of sine functions. These are a graphical symbol for the waves that can, in certain contexts, represent a particle.
As 'a' is a horizontal distance, it represents the amount of Euclidean space to an appropriate scale that the particle occupies and not the amount of space that it would occupy if the curvature were unfolded or smoothed out. This latter quantity is represented by the height of the point above the base line.

An analogy is a knot in a piece of string. This could, indeed, serve as an alternative graphical symbol; it could be called a 'knot symbol'. The distance across the knot along a straight hne between two points on opposite sides of the knot is quite short. But when one follows the twists and turns of the string between the same two points one measures a much greater distance. A knot can be described, a little paradoxically, as a long piece of string that occupies a short distance. The point symbol for a particle represents the notion symbolically that the particle is a large amount of curved space in a small volume.

Let us imagine a person who studies a knot but can only measure distance along a straight line. He cannot get inside the knot and measure the length of string that is there. He can only measure the distance along a straight line that is taken up by the knot. In consequence his observations and measurement convince him there there is only a small length of string. It is only when the knot has been untied that he discovers the truth.

All analogies are imperfect, but this one serves to illustrate that it may mean something to say that a particle contains more volume than can be observed or measured. We approach the particle from the outside, from Euclidean space, and can ascertain thereby only how much Euclidean space there is between points on opposite sides of the particle. By this process we are unable to discover how much curved space there is between the same two points.

The value of a point symbol is enhanced if one adopts the convention that a tilt of the axis from the vertical represents an acceleration. This is shown in Fig. 7B. The shape of the graph is distorted by the tilt, which is a symbolic way of representing the fact that something has been done to the particle when it has been accelerated in flat space. Such a distortion can be thought of as occurring when the particle is hit by another one.

There is one particular feature about both a knot and a point symbol that might represent actuality but is more likely to be misleading. This is the suggestion conveyed by both symbols that space has dimensions additional to the three that we attribute to it. A piece of string has a two-dimensional cross section and is knotted into a third dimension. Hence a knot symbol suggests a space with one additional dimension. In the point symbol a horizontal distance represents volume in three spatial dimensions, each at right angles to the other two. By the same convention a vertical distance represents volume in three further dimensions, each at right angles to the other two and also at right angles to each of the three dimensions represented by a horizontal distance. The point symbol represents a six-dimensional space. Does this correspond to actuality?

It does not seem to. True, one cannot represent curvature in the imagination without picturing curvature into something. A line has one dimension and it curves into a second one; a surface has two dimensions and curves into a third. When one tries to picture transverse electromagnetic waves one is led to think of the electromagnetic field as curving into a fourth, and perhaps also a fifth dimension. The urge to interpret basic physical phenomena with the help of additional dimensions was not new at the beginning of this century. But Einstein made it more insistent when he introduced the notion of curved space. One conjured up extra dimensions for space to curve into in a desperate effort at representational understanding. But one should not forget that such efforts are misplaced.

Let me put what I am trying to say in a different way. Anyone who viewed the point diagram from below a line in the plane of the paper would see only the length 'a' and not the point. This length symbolizes a three-dimensional volume. But the whole diagram symbolizes a six- dimensional volume. To assume that actual space is six-dimensional would get us out of some difficulties. But I distrust easy ways out of difficulties and, besides, this one is not new and would have been taken years ago by those who are seeking a unified field theory if it were sound.

In this instance prepositions need to be carefully noted. Einstein did not speak of curvature of a three-dimensional body in space; he spoke of curvature of space. This is something of which the imagination cannot, and should not, hope to form a picture. Here again I make no apology for introducing some metaphysics. If an apology is owing it is for my inability to introduce more. More is needed in basic physics if understanding is not to lag so far behind knowledge that both will be lost. But I must return to the point symbol.

Rightly or wrongly this point symbol as shown in Fig. 7B represents the notion that there is an equilibrium state of curvature, for which there is some sort of symmetry. Disturbance of the symmetry is resisted and, on removal of the disturbing force, the previous condition is restored; Fig. 7a returns to Fig. 7A with cessation of acceleration. To suggest this is to attribute elasticity to an elementary particle, and there are reasons for thinking that one is justified in doing so.

The symbol would aptly represent the notion that a finite, though very minute, interval of time elapses between the moment when the distorting force has been removed and symmetry is restored. The graph is not a bad symbol for a particle that shivers like a jelly after it has been hit and in which the distortion can be accentuated if repeated impacts have a certain, resonant, frequency. Whether this is a wanted symbol or not, I should not like to say categorically. But resonance is a known phenomenon in nuclear physics and so this feature of the symbol may prove useful.

We are in the habit of thinking of inertia as the property by virtue of which a body in flat space resists a change in its velocity. Fig. 7B does not quite symbolize this notion. What it does symbolize is that inertia is an indirect effect. The immediate resistance of a particle that is subjected to a force is resistance to its distortion. The acceleration in turn is the consequence of this distortion, of the tilting of the axis. Here again, I think that the hint ought to be taken. It would be worthwhile to inquire whether the acceleration imparted to a particle by an impact is the direct effect that we picture or the indirect effect of a change in the way the curvature of the space constituting the particle has been altered by the impact.

Fig. 7C is a point symbol for a particle that is in a gravitational field and free from restraint. The tilt of the axis from the vertical again represents the acceleration of the particle. The lack of distortion represents freedom from restraint. The angle of the base of the graph to the horizontal represents the potential gradient in which the particle finds itself.

It should be noticed that the acceleration appears as such to an observer whose frame of reference is parallel to the edges of the paper, but an observer whose frame of reference was parallel to the base line of Fig. 7C would not think that the particle was being accelerated. He would be in the position of a person who is falling freely and observes a stone that is doing the same.

Fig. 7D represents a particle in a gravitational field when the particle is prevented from falling by a restraining force. The distortion symbolizes the restraint and the vertical position of the axis symbolizes the fact that there is no acceleration.

Fig. 7E is the graphical symbol for elastic collision between two particles of equal size. It shows the distortion that results for each from the collision. It is a measure of the force of impact between the particles. The opposing tilts represent the accelerations with which the particles rebound from each other after impact.

The point symbol, it has already been shown, represents elasticity, not as a property of a 'particle stuff', but as a property of curvature. By this convention the same symbol also represents the identity of mass and energy that was discovered by Einstein. When the curvature is increased the result is an increase of mass. But space resists a change in its curvature from a condition of symmetry. What is done to overcome this resistance is called a supply of energy.

What can be expressed somewhat vaguely and in qualitative terms only by graphical symbols can be expressed precisely and in quantitative terms by letter symbols, which are the basis of algebra. I am now suggesting that there is profit in taking the notion quite literally that a particle is a region of very highly curved space. Einstein took the notion quite literally that a field of force is such a region. He developed the notion with the help of letter symbols, and with most fruitful results. If the logic of algebra is applied with similar rigour to the curved space called a particle the result will be, I venture to suggest, equally fruitful.

Point Symbol for a Composite Nucleus
The picture that most of us form of a composite atomic nucleus is rather like a blackberry. We think of a number of minute, pip-like, spheres packed tightly together. Some of these represent neutrons, others protons. To distinguish them from each other we probably endow each kind in our imagination with a distinctive colour. If I were to give a name to this picture, I should call it a 'berry symbol'. I should certainly not call it a model. I doubt whether any nuclear physicist would claim that it was a true model of actuality.

A berry symbol has its uses, but it can also be sadly misleading. Let us consider some of its implications.

Each of the pips, be it a proton or a neutron and collectively called a nucleon, has mass. According to the traditional relativistic theory of gravitation it is surrounded by the faint halo of a gravitational field to which I have alluded already. The gravitational forces within the nucleus thus appear to an adherent of the traditional school as very faint indeed. The berry symbol represents symbolically a structure in which these gravitational forces provide a negligible amount of cohesion. According to the new theory of gravitation, be it noted by the way, the nucleon does not have even this faint halo of a gravitational field.

The same symbol shows the protons as very close together, as almost touching. But as these repel each other according to the inverse square law, the forces of repulsion between them are represented in this symbol as enormous. It is a symbol for a structure that contains little else but strong disruptive forces.

Accepting the implications of this symbol, some nuclear physicists have had resort to the ad hoc hypothesis of a new kind of force in nature, which is postulated as an addition to the gravitational and the electro-magnetic ones. The new kind of force is assumed in this rather makeshift hypothesis to operate with great intensity over very small distances only and to be one of attraction between nucleons. Its sole justification is that it is the best that has been found consistent with the traditional hypotheses about gravitation and the nature of a particle. Others seem to have adopted the ad hoc hypothesis that has been mentioned in Appendix F, namely that matter is permitted to disobey the inverse square law over very short distances.

I venture to suggest that neither the hypothetical nuclear force nor the hypothetical dispensation from the inverse square law would ever have been invented were it not for the misleading character of the berry symbol. I should like to show below that we have been mistaken in thinking that gravitational forces within the nucleus are very weak and electrostatic ones enormously stronger. There are good reasons for the conclusion that the gravitational forces greatly predominate in a nucleus of low atomic number and that the two kinds of force are of the same order of magnitude in a nucleus of high atomic number. Let me show why.

I have already drawn attention to the fact that electric charges behave in the same way in gravitationally curved space as they do in flat space. They are quite unaffected by the gravitational kind of curvature. Let me illustrate this by an analogy. A trumpet is a curved tube. But the curvature is introduced mainly for the convenience of playing. If the tube were straight, the trumpeter could not reach the stops. The pitch of the note that he blows is practically the same as it would be if the trumpet were not curved.

In this tube two nodes of a standing wave are separated by a certain distance. What is it? If the tube is bent back on itself, the nodes may be quite close to each other along a straight line. But a distance along this straight line is irrelevant. It does not affect the pitch. The curved distance measured along the tube is the one that counts for the determination of acoustic results. In this illustration the trumpet is shortened in such a way as to bring the stops within reach of the trumpeter's hands, and yet this shortening does not reduce the distance between nodes in a standing wave.

In curved space, when the curvature is of the gravitational kind, electric charges behave like standing waves in a curved trumpet. The curvature does not affect the separation between them. So the question arises whether the charges in a composite nucleus are all in such gravitationally curved space. If so, the nucleus could be represented by a knot symbol as shown in Fig. 8 for a helium nucleus. Each U-shaped element represents symbolically the quantity of curved space that is occupied by the mass of a nucleon, be it a neutron or a proton. The two nuclear charges that characterize helium are represented by plus signs. The volume that external measurement assigns to the nucleus is symbolized by the horizontal distance across the diagram. The volume of gravitationally curved space that constitutes the nucleus (perhaps six cubic centimetres) is symbolized by the labyrinth measured from end to end. To be a quantitative symbol the U-shaped elements would have to be very long and thin.

Fig 8 A possible graphical symbol
for a helium nucleus

This knot symbol represents the notion that the nucleons are all fused into the same curved space. The berry symbol does not do so. It represents the notion that each nucleon preserves its identity. Which is the true notion?

So long as one thinks of an elementary particle as a hard, round, unbreakable sphere one will prefer the berry symbol. One will then have no choice but to invent nuclear forces or something else to explain the cohesion of the nucleus. But if one casts one's mind back to the early days of relativity and remembers the conclusion that a particle is curvature, one will be more inclined to accept the notion that when two curvatures come into very close proximity they fuse into one. The knot symbol will then appear the more appropriate.

In this there is no need to invent nuclear forces. The distance between positive charges has to be measured along the labyrinth, for charge ignores gravitational curvature. This distance separates the charges by a substantial amount. According to the inverse square law the force of repulsion between them is moderate. But all the nucleons are in a region where perhaps six cubic centimetres of Euclidean space are curved into the small volume of the nucleus. The curvature is very steep, fantastically so in comparison with the curvatures in the free gravitational fields with which we are more familiar. The cohesive forces of gravitation between the nucleons are thus enormous.

A point symbol illustrates the same features. One is shown in Fig. 9 for a boron nucleus.

Fig 9 Point symbol for a
boron nucleus

The horizontal distance along the base symbolizes the volume as generally stated, while the height of the point symbolizes the volume as it appears to the five charges. If the symbol were drawn to scale its height would be very great. It would then be clear that the charges were well separated from each other.

A few further implications of the point symbol are worth exploring. Every charge within the nucleus repels every other one. So each charge is subjected to a bigger force of repulsion if there are many other charges than if there are few. If charge density is defined as the number of charges per unit volume of the space that is curved or folded into the nucleus, it is easy to show that for constant charge density the force of repulsion on each charge increases with the number of charges present. Now constant charge density is obtained when the ratio of protons to nucleons is constant. So the tendency for the nucleus to disrupt under the influence of internal electrostatic forces would increase with increasing atomic number if the number of neutrons always bore a constant ratio to the number of protons. For the nucleus to have coherence one should expect this ratio to increase with the atomic number. Observation shows that it is so. The larger the atomic number the greater is the relative number of neutrons in the nucleus. This ensures that the charge density decreases with in- creasing atomic number and that the separation between adjacent charges increases also, giving thereby less force of repulsion between one charge and the others.

Let us return to Fig. 7E. It represents a collision between two particles of equal mass. If one of them is larger than the other and carries positive charges while the smaller one is a neutron, one has Fig. 10. The tilt of the neutron, and therefore its acceleration, is shown as much greater than that of the heavier particle; as it is in actuality.

Fig 10 Collision between neutron and nucleus

The point symbol allows for other consequences of a collision between particles. This is, it must be remembered, not between things made of 'particle stuff' but between things made of 'curvature', if the expression be allowed. The first result of the collision is a distortion of each colliding particle. It is not difficult to believe that this distortion can be of various kinds, depending on the violence of the collision and, perhaps, on the length of time during which the colliding particles are close to each other. If the collision is comparatively gentle, it may cause elastic rebound. If it is more severe, it may lead to fusion of the colliding particles. This is called capture. When the nucleus is so large that the forces of repulsion almost predominate over the forces of cohesion, the collision may cause breakage, i.e., radio-activity and the ejection of particles.

This latter event can perhaps be regarded as a consequence of disturbing the relative positions of the charges in the nucleus. At rest, it is to be assumed, these take up positions in their many cubic centimetres of highly curved space where they are in equilibrium. But the distortion that immediately results from a collision displaces the charges so that some come closer together while others are more widely separated than before. Those that approach more closely then repel each other strongly enough for one or more to be ejected. Here the resonance that I have already hinted at (and that would be a consequence of the time lag between the production of a distortion and the regaining of the previous symmetry) may help to explain sundry phenomena in nuclear physics. But I have to resist the temptation of further exploring the ways in which a point symbol can suggest explanations for the observed action of particle on particle.

I do, however, feel under an obligation to say something about question (7) of those listed on page 273. It is the question that was discussed more fully in Appendix G, namely, why an extinction can occur without both producing radio-active effects and leaving a residue of the nucleus of an atom with lower atomic number.

The question has answered itself in the course of this inquiry into more important aspects of nuclear physics. If it is true that a particle is curvature, that the components of the composite particle called a nucleus share a common curvature, and that the extinction of a particle is really the conversion of bound to free curvature, there is no difficulty. The nucleus is, regarded as a region of bound curvature, a single unit. If and when it collapses into free curvature, it does so as a whole. The extinction of some part of the nucleus can therefore never occur. It is all or nothing.

This conclusion raises the further question whether the probabilities of the extinction of positive and negative charge are necessarily exactly equal. If not an insulated body must acquire a small charge with time which might have observable consequences for a body of the size of the earth. There is, I suggest, a further field for research.

In this final appendix I have tried to give a fair sample of the difficulties that one encounters in basic physics. I have given prominence to those difficulties that are associated with various existing hypotheses about the nature of a particle and have mentioned difficulties that are inherent in my own theories as well as in those of others. I have tried to maintain a correct sense of proportion and to convey a true picture of the degree of tentative- ness or assurance that I feel about the various conclusions that I have presented.

Many more questions remain and as I write this it almost seems as though they were persons. Some tell me of fascinating avenues that they invite me to explore. Some remind me that I have already gone quite a little way towards providing them with some sort of an answer and urge me to proceed further. Some are baffling, some tiresomely clamorous, Some look at me a little spitefully and say that, if I do not satisfy them, they will prove that all my theories are invalid. But I must let them clamour and hustle. If I yield to these questions there will only be new ones. As I have said already, science is hydra-headed.

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