by     Reginald O. Kapp


Appendix B - Can a Galaxy Acquire an Infinite Mass?

B.l: Relative Growth of Neighbouring Galaxies
Let us consider the boundary around a galaxy that has been called the reversal zone in Chapter 10. This, it will be remembered, is a surface that entirely surrounds the galaxy and at which the potential gradient is zero. New matter that originates within the reversal zone falls towards the galaxy enclosed by it. New matter that originates beyond this zone falls towards a neighbouring galaxy.

The position of a reversal zone anywhere is a complicated function of the distribution of masses within a volume large enough to comprise many galaxies. But let us, as a first approach, consider two neighbouring galaxies with their respective reversal zones. The volumes within these zones have been called the boundaries of the galaxies. The total mass within each domain acts as though it were at the centre of gravity of the domain. Let the total masses within two neighbouring domains be, respectively, m1 and m2, where m2 is the larger.

According to the hypothesis of continuous origin the gross income of a domain is directly proportional to its volume. The volume, in turn, is a direct function of the mass. So the larger, and more massive, domain has the greater income of newly originating matter. If the average rate of origins within both domains exceeds the average rate of extinctions the larger domain will increase in mass and volume more rapidly than the smaller one. The ratio m1 / m2, will become a diminishing fraction. If the respective volumes are V1 and V2 and the respective distances from the centres of gravity to the reversal zone are D1 and D2, V2 will gain on V1 at the same time as m2 gains on m1, and the greater the discrepancy between V2 and V1 becomes the more readily will m2 gain relatively to m1. The ratios m1 / m2, and V1 / V2 will tend at an exponential rate towards zero. The same holds for the ratio D1 / D2 In other words the boundary between any two galaxies in domains that contain unequal masses is continuously being pushed towards the centre of the smaller one so long as origins exceed extinctions on both.

This must happen whatever the difference in their masses is, provided only that there be a difference. Even if this is very slight to begin with, the time must inevitably come when the boundary will reach the edge of the smaller galaxy. It will not, however, stop there. It will encroach on the substance of the smaller galaxy and leave this behind it and within the domain of the larger one. Parts of the smaller galaxy will then begin to fall away from it. The process will not even stop when the centre has been reached. It will continue until its whole substance has become forfeit to its more massive neighbour.

For the model based on Asymmetrical Impermanence the rate of origins always exceeds the rate of extinctions, for the latter rate is zero. Hence for this model a domain that is only slightly more massive than its neighbours must always compete successfully with them for hydrogen. It must grow at their expense and thereby become even more massive. In doing so it must also become ever more capable of competing successfully. In the course of time it must swallow every other galaxy in turn. The model based on Asymmetrical Impermanence is thus one that finally consists of but one single concentration of matter. If continuous origin has been going on for all time this model will have an infinite mass and exercise an infinite gravitational pull on new matter. It is not a model that conforms to actuality.

B.2: Wrong Use of the Concept of Infinity
Suppose that one could get round the objection to Asymmetrical Impermanence that has just been mentioned. It would not be difficult, for one can always get round an objection to a favoured theory with the help of suitable ad hoc hypotheses.

Here the choice might fall on the hypothesis that the inverse square law for force receives a correction such as to limit the distance at which gravitation has a finite value. Galaxies that were further from the biggest galaxy than the limiting distance would then be beyond its range and would not be swallowed up. Thus one would be able to explain the observed existence of discrete galaxies. But even so, this way out of the dilemma could not be allowed, as the following consideration shows.

According to Asymmetrical Impermanence, even when coupled with the further hypothesis that gravitational forces act only over finite distances, the size of every galaxy would be a direct function of its age. Young ones would be small and old ones would grow continuously without limit. Those that had begun an infinite time ago would have become infinitely massive. The potential gradient around them would be infinitely steep. Particles of surrounding hydrogen would be falling on to them with an infinite acceleration. Any atoms of new hydrogen that originated within a finite distance from such galaxies would instantly acquire the velocity of light and become infinitely massive.

The above conclusions are alone surely reason enough for rejecting the hypothesis of Asymmetrical Impermanence. But the reason needs to be explained, for it does not always seem to be properly appreciated. I have pointed out already that a cosmological model can be accepted only if it resembles actuality, and supporters of Asymmetrical Impermanence have claimed, rightly or wrongly, that a model that implies nebulae of infinite mass does not necessarily fail to resemble actuality. All that we can know about the actual universe is, they point out, what is within the optical horizon and therefore observable. The probability that one of the infinitely massive nebulae would occur within this horizon is, it has been said, extremely small. That none are observed does not, therefore, prove that none exist. This was, for instance, the point made by one of the supporters of Asymmetrical Impermanence, F. Hoyle, in a correspondence with myself conducted in the pages of Nature.1

The argument is of doubtful validity. It holds only if one can postulate that the infinitely massive nebulae are all located at some distance from us such that their infinitely powerful fields at an infinite distance are undetectable here and now. Presumably, then, their distance is some higher power of infinity! However, this is not the only or even basic objection. One must reject a model that includes infinite physical quantities, not because it can be proved wrong by observation, but because it is conceptually wrong. Those who accept this model with equanimity misunderstand the meaning of the concept infinity.

When the symbol for infinity occurs in a mathematical expression it can be translated into words as 'an indefinite and unspecified number larger than would define any physical reality with which this expression is concerned'. The words in italics are essential. To say that there can be any time and place where there is a physical quantity for which this symbol represents the correct measure is to misread the symbol.

The meaning of other mathematical symbols is that, when they are replaced by numbers, these numbers will exactly define the quantities that the symbols represent. The numbers will be neither too large nor too small. To imply that the symbol for infinity is similar and can be replaced by a number that is neither too large nor too small is to assume that the symbol stands for a very large but nameable number. Laymen make this mistake every day, but scientists have to learn to avoid it.

B.3: Growth of Domains for the Model Based on Symmetrical Impermanence
Thus the model based on Asymmetrical Impermanence must be rejected on conceptual, as well as on observational, grounds. Does the model based on Symmetrical Impermanence fare any better?

I have to admit that I am not sure. If it could be proved that this model is not self-adjusting in the sense of automatically limiting the size of galaxies to finite values I should reject it at the cost of sacrificing the great explanatory power that has been demonstrated for this hypothesis, both in the main parts of this book and in the appendices that are to follow. Only a searching mathematical study can show whether it is so and this study is among the many that I prefer to leave to others. I shall rest content to show here how a superficial investigation is encouraging and suggests that Symmetrical Impermanence is likely to stand this test, as it has stood all the others to which I have subjected it.

If the size of a galaxy is to be self-adjusting extinctions within its domain must equal, or exceed, origins when the galaxy has reached a certain size. This can only happen when the average mass density in the whole domain equals or exceeds the equilibrium value. The quantity dm / dt for the domain will then be negative.

This would happen if the reversal zone around an old and massive galaxy were, from time to time, to be pushed inwards by the new galaxies that grow around it. Such a process would transfer volume from the domain of the old and more massive galaxies to the domains of the newer and less massive ones. As much of the mass within a domain is concentrated in the galaxy at its centre the loss of volume of the older domain would not be accompanied by a proportional loss of mass, so the mass density would increase. If it did so until the equilibrium density was exceeded the galaxy would dwindle.

Let the average mass densities in two neighbouring domains be, respectively, σ1 and σ2. If the model is to be self-adjusting the ratio σ1 / σ2 must be a direct function of m1 / m2 or V1 / V2. For then the more massive and more voluminous domain would have the greater mass density. Its loss by extinction would be relatively the greater. If it is so the reversal zone will move at times away from the incipient and towards the older galaxy until extinctions have brought the density to below the equilibrium value.

One can only know if this happens after one has calculated the way in which the gravitational fields of a large assembly of galaxies are superimposed. In other words the contours of the astronomical landscape and the changes that it undergoes must be known with some precision. What is known at present is too vague to permit the question to be answered with any certainty. But one may approach an answer by a process of successive approximations. Suppose one begins by ignoring the fields of all galaxies except two, an incipient one and one of its older neighbours. One will then define the position of the reversal zone between them by the approximate equations given in Chapter 10.

(D1 / D2)2 = m1 / m2 . (10a)

If one puts as a further approximation,

V1 / V2 = (D1 / D2)3

one can write

V1 / V2 = (m1 / m2)3/2

from which

σ1 / σ2 = (m2 / m1) 1/2 . (Ba)

If this were accurate the larger mass would be in a domain with the smaller mass density. The rate of extinctions per unit volume would be smaller in the domains with the smaller volume. The model would be the reverse of self-stabilizing. But equation (lOa) cannot be very near the truth.

One approximates more closely to the truth when one takes a third galaxy into account. This has been done in Chapter 12 with equations (12c) and (12f). But these two are not quite correct. A further approximation has been adopted there by substituting an astronomical pass for a summit.

The potential gradient near the pass is approximately

E2 = -4Gm2r / D3

derived from (12f)
where D is the distance from the reversal zone to the older galaxy with mass m2 and r is the distance from the reversal zone to the astronomical summit before a new cloud has begun to form there. Let this have formed and have acquired mass m1. The potential gradient attributable to m1 only is then

E = Gm1 / r2

At the reversal zone the two gradients sum to zero and one can write

Gm1 / r2 = 4Gm2r / D3

from which

m1 / m2 = 4( r / D )3

The ratio of densities is then

σ1 / σ2 = 4

The further approximation gives four times the density of the older domain to that of the incipient cloud so long as r is small. If this were the true relation the density in the domain of the older galaxy would be one quarter of the equilibrium value when this value was reached by the incipient cloud. The older galaxy would continue to grow more massive and would extinguish the incipient one. The objection already found against Asymmetrical Impermanence would also be valid against a Symmetrical Impermanence.

However, equation (12f) is still misleading, if not as much so as equation (lOa). As has been pointed out at the end of Chapter 12, an equation that gives the potential gradient around an astronomical summit, as distinct from one at an astronomical pass should take some such form as

E = - { GmDr / (D2 - r2 )2}φ(r / D).. ...... (12g)

By this nearer approximation the position of the reversal zone may perhaps be defined by

Gm1 / r2 = -{Gm2Dr / (D2 - r2} φ ( r / D)

from which

m1 / m2 = - { D r/( D2 - r2 )2φ (r / D)

If r is very small compared with D this gives

σ1 / σ2 = -φ( r / D ) ........... (Bb)

As an astronomical summit is flatter than a pass the function varies directly with r/D, as was said in Chapter 12. Hence it is more than probable that the function is such as to cause the smaller domain also to have the lower mass density. If so the possibility need not be excluded out of hand that the mass density in the larger and older domain may exceed the equilibrium value while that in the domain of the new galaxy is below it.

It is, however, not sufficient for φ(r/D)to be the right kind of function in order that the size of galaxies may always be finite. It is also necessary for the half-life of matter to be below a certain value. If the half-life of matter were infinite and matter were originating continuously as advocates of Asymmetrical Impermanence assert the whole of the material universe would form a single infinitely massive concentration whatever form was taken by φ(r/D). What the half-life must be depends on the nature of φ( r / D), but some general considerations will be given in Appendix C, which show that, by astronomical standards, the half-life must be less than one might expect. Further considerations to be given in later appendices will, however, show that a rather short half-life is consistent with sundry well-known facts of observation. Astrophysicists, geologists and biologists will indeed have reason to welcome a rather short half-life. For it helps to explain several facts that have hitherto defied explanation.

The conclusion that is hinted at by equations (Ba) and (Bb) can be expressed differently as follows. If there were only two galaxies the larger one would inevitably prevail by competing successfully with the smaller one. But with a three-dimensional arrangement the time comes when an old and large galaxy is surrounded on all sides by small new ones. It is suggested that their combined competitive power suffices for their domains to encroach successfully on the older one. If the half-life of matter is short enough the encroachment suffices to increase the mass density of the large galaxy to above the equilibrium value, leading thereby to a reduction in its mass.

B.4: The History of a Domain
Let us consider, in view of the automatic adjustment that seems to follow from equation (Bb), how the mass of a domain must vary with time. There will be a moment when neighbouring domains contain equal masses. Their average mass densities and volumes must then also be equal and the astronomical pass between them must be equidistant from their respective centres.

In expanding space the centres drift apart and the domains grow more and more voluminous. Therewith they receive new hydrogen at an increasing rate and become more and more massive. The new hydrogen falls on to the galaxies at the centre of the domains and is added to their masses. But only when the hydrogen completes the journey. Some of it must, according to continuous extinction, become extinct while it is under way. The proportion that does so must increase with an increasing average journey, and so an increase in volume of a domain does not result in a proportional rate of increase of the mass of the galaxy at its centre. But, nevertheless, the rate at which this galaxy grows by capture does vary directly with the volume of the domain.

This volume does not grow indefinitely. The time must come when the potential gradient around the neighbouring astronomical summits falls to the value at which a new cloud can begin to form. When this happens the new cloud competes with the surrounding galaxies for hydrogen. In other words its domain encroaches increasingly on theirs. The older domains then begin to dwindle in volume and mass. But the regions that they lose are the outermost ones and these are very tenuous; the dense central core remains in the old domain. Hence the average mass density of the old domain is greater than corresponds to its size and begins to be reduced by a preponderance of extinctions over origins. The adjustment takes time and so there is a time lag, a phase displacement, between the process of reducing the volume and of reducing the mass. But this need not concern us at the moment.

Eventually the new cloud becomes a new galaxy, as massive as its neighbours. It and they drift apart. Their domains grow and therewith their incomes. They begin to grow again.

It is not difficult to understand that a new cloud must begin to form on an astronomical summit as soon as the gradient has fallen to the critical value, and that this must happen in time for every astronomical summit in extragalactic space. For the gradient continues to jail while space expands and the neighbouring galaxies drift further away.

Clearly the average distance between two neighbouring galaxies at the moment when a cloud begins to form must have a constant value Dav and the average distance of the incipient cloud from the two galaxies is half this value or Dav / 2. When the cloud has itself become a galaxy and its distance from the neighbours has grown from Dav / 2 to Dav a new cloud will begin to form on the new astronomical summit that has developed.

Thus the time interval between successive generations of galaxies must average the time taken for the linear dimensions of space to double themselves. This has been given as three-and-a-half thousand million years. But it depends on the value of Hubble's constant, about which there is some uncertainty at the time of writing. This time is the period of fluctuation in the mass of a galaxy.

1. Nature, Vol. 165, pp. 68 and 687, on respectively 14 January and 29 April, 1950.

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