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) xr*^{3},
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 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.

*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.

*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.