TOWARDS A UNIFIED COSMOLOGY

by     Reginald O. Kapp

PART V - APPENDICES

Appendix C - The Half-life of Matter


C.1 : The Need for Quantitative Thinking
The reasoning presented in this book has purposely been predominantly qualitative and this because, as was explained in Chapter 12, one cannot usefully do much quantitative work on any problem until one has reached clarity about the concepts concerned. A thorough understanding of the meaning of the algebraic symbols that are to be used must precede mathematical treatment if this is to be profitable. If mathematics is applied prematurely, it becomes no more than an idle exercise; it remains unrevealing; it may mislead.

But, nevertheless, quantitative work should, where possible, always follow the search for meaning. That quantitative thinking should take second place in order of procedure does not imply that it necessarily takes second place in order of importance. For this reason a small amount of mathematics has proved to be unavoidable. Its result has been reassuring. But much more will have to be done in order to establish the Principle of Minimum Assumption as a sound basis.

Further tests of this principle, be they quantitative or qualitative, will involve many disciplines other than astronomy; any branch of physics may be able to contribute something; a great variety of different mathematical methods may have to be employed; a range of knowledge, skill and experience will be required that cannot be possessed by any one scientist. So I have to rest content to leave the task of further testing and development to others. But before doing this I should like to draw particular attention to the importance of the cosmic constant that I have called the half-life of matter.

According to the traditional hypothesis (Bl), at the beginning of Chapter 3, an elementary component of the material universe never becomes extinct. At most it is converted from one form into another. This is equivalent to saying that the half-life of matter is infinite. But according to the Hypothesis of Symmetrical Impermanence the half-life of matter is finite. It was mainly quantitative considerations that caused me to reduce the half-life from infinity to an unspecified finite value. Before Symmetrical Impermanence can become established, further quantitative considerations must bring this value between definable lower and upper limits.

It appears that such limits can be arrived at by several distinct lines of reasoning, each starting from different known facts. If the Hypothesis of Symmetrical Impermanence is valid, all these limits must leave a bracket of possible values. The lower limit set by one approach must not be higher than the upper limit set by another. Every effort must be made, moreover, to narrow the gap between the lowest and the highest of the possible values. Let me give a hint here as to ways in which these limits may be estimated.

C.2: List of Symbols

Na =

Average number of elementary components per unit volume in a fair sample of the universe. (A fair sample may have to be extensive and to include many extragalactic nebulae; for the mass density in the domain of any particular nebula may differ significantly from the average value.)

Neq =

Number of elementary components per unit volume at the equilibrium density. (This is the density at which the rates of origins and extinctions per unit volume are equal.)

N0 =

Number of elementary components that originate in unit volume and time.

ne =

Fraction of elementary components in any volume that become extinct in unit time.

Ts =

Time during which the linear dimensions of space double.

Tm =

Half-life of matter.

p =

Tm / Ts

q =

Neq / Na

m =

Mass of the earth at any time t

m0 =

Mass of the earth when t = 0

C.3: The Earth's Loss of Mass.
According to Symmetrical Impermanence every body is losing mass at a rate determined by the half-life of matter. But bodies like the earth, the moon and the sun are also gaining mass by capturing it from their surroundings. However, the sun must compete successfully with the earth and the moon for any matter that is in process of falling towards the solar system. Such matter must nearly always fall past the planets and their satellites and on to the sun.

Hence it would need some research before one could say whether, on balance, the sun is gaining or losing mass, whether dm / dt is positive or negative. But one may feel sure that the earth is gaining only a negligible quantity and that, according to Symmetrical Impermanence, dm / dt must be negative and determined to a very close approximation by the half-life of matter.

This can be expressed by the equation

dm / dt = -ne m

from which

m = m0e – net

so

m0 = menet

and

ln(m0 / m) = net

For the half-life of matter

t= Tm and ( m0 / m ) = 2

Hence

ln2 = 0.69 = neTm …………. (Ca)

It follows that for any values of m0 and t

ln( m0 / m )= 0.69t / Tm………. (Cb)

For small values of t/Tm this can be replaced by

(m0 / m) = 1 + 0.69 t / Tm ………. (Cc)

From this equation it is seen that the loss of mass of the earth during the last 1,500 years has been one part in ten million if Tm is 1010, one part in a million if Tm is 109 and one part in a hundred thousand if Tm is 108.

The value of g has diminished correspondingly. Any calculations that have assumed g to have remained constant during 1,500 years must therefore be in error by a corresponding small fraction.

Eclipses are among events affected by such calculations. If the earth was more massive in the past than it is now, it had also a firmer hold on the moon. This body was nearer to the earth and completed its orbit in a shorter time than it would have done if g had been no greater than it is today. If the records of past eclipses were completely accurate, and if all other quantities that enter into the calculation of an eclipse were known with absolute precision, one could compare recorded times and places with calculated ones. If the calculation assumed that g had remained constant, any difference between records and the result of calculations would then reveal the change that g had undergone.

But records cannot have been so very precise 1,500 years ago and calculations made today have to allow for a small and unavoidable margin of error. So a small change in the value of g would be masked by various uncertainties. Among the circumstances that significantly affect the time and place of an eclipse tides are important. They cause the earth to rotate more slowly while loss of mass by extinction causes it to rotate faster. For this reason a rather large change in the value of g would be necessary for a comparison of recorded and calculated eclipses to reveal it.

Only experts in astronomical measurement could tell us how great a rate of change of g could be detected with the help of all available data. If no such change has in fact been detected, one may safely conclude that the rate of change is less than this value, though one cannot know how much less. The negative evidence that no change has been detected would thus set a lower limit to the half-life of matter. It seems probable that a detectable rate would have to be much greater than corresponds to a loss of mass of one part in a million during 1,500 years, and so the lower limit set by this consideration is probably much less than 109 years.

A small change during 1,500 years is, however, equivalent to a large one during the whole of the earth's existence and it may be possible to find reasons why the earth cannot have been more massive at its beginning than some definable value. I do not know of any limiting consideration and yet I do not like to contemplate the notion that the earth has ever been many times as massive as it is today. Probably my disinclination to do so is emotional and therefore inadmissible in a scientific inquiry. So let any objection to a large value for the initial mass of the earth be ignored unless it can be more rationally justified.

At the time of writing this the estimate for the age of the earth that is given by the radium clock and seems to have the widest support is 2.6x109 years. Reasons for reducing this estimate greatly will appear later, but if one gives this value to t in equation (Cb) one obtains the following table.
In this m0 / m is the ratio of the earth's initial to its present mass and Tm is the half-life of matter:

Table 1

Tm x 109 years
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

m0 / m
395
89
36
19
12
9
7
6

It is doubtful whether anyone knows anything at all about the mass of the earth, its condition or its other properties at the time when it began. So any figures concerning its mass at that time can neither be proved true or false.

Knowledge begins at a date that may roughly be put at two thousand million years ago and so it is relevant to consider what the maximum possible value of the mass of the earth can have been at that time. If the half-life of matter is 109 years, its mass two thousand million years ago was about four times its present mass. Its radius was therefore about 60 per cent greater than it is now. But reasons will be found in this appendix, as well as in Appendices D and E, why this is probably a gross under-estimate. Any emotional distaste one may experience for the notion that the earth has today but a small fraction of its initial mass must yield to some rather cogent facts, which are partly astronomical, partly geological and partly biological.

C.4: The Rate of Extinctions per Second
According to the new theory of gravitation that has been presented here, the gravitational field is quantized. The gravitational force occurs in jerks. But these are not observed; the force seems to be continuous, not only for the attraction exercised by the whole earth but also for that exercised by those leaden spheres that have been used in the measurements of the gravitational constant G by Cavendish, Boys and Poynting. It is evident that the jerks follow each other in quick succession and as each jerk is the consequence of an extinction the rate of extinctions in a sizeable lump of lead must be high. This consideration sets an upper limit to the value of Tm.

From equation (Ca) the fraction of a given mass that becomes extinct in unit time is 0.69Tm. From this it is easy to find that the fraction that becomes extinct per second is 2.2 X 10-8 / Tm if Tm is measured in years. A gramme has the mass of about 6x1023 protons and if the elementary component that becomes extinct has the mass of a proton the number of components becoming extinct in a gramme during every second is 1.3 X 1016 / Tm. If Tm is 109 the number of extinctions in one second is over ten million, much more than sufficient to make the field appear to be continuous. It follows that a mass of one ton of hydrogen would produce quanta of gravitation at a rate of over one per second if the half-life of matter were 1024 years. It is evident that a very long half-life is consistent with the apparent continuity of the gravitational force.

C.5: The Equilibrium Density
During the time Ts that it takes for the linear dimensions of space to double the volume of space increases eightfold. If the average density Na is to remain constant there will be seven new elementary components at the end of time Ts, for every one that there was at the beginning of that time.

During the time Ts, the number of elementary components becoming extinct is neNaTs, and so the gross number of origins is

N0Ts = 7Na + neNaTs ……….. (Cd)

From this it follows that in a fair sample of the universe the ratio of origins to extinctions during unit time is

( N0 / neNa = ( 7 / neTs ) + 1

Substituting for ne on the right hand side from equation (Ca) one obtains

( N0 / neNa ) = (7Tm / 0.69Ts ) + l

Putting Tm / Ts = p then gives

( N0 / neNa ) = 10p + l nearly.

Now at the equilibrium density, Neq the rates of origins and extinctions are equal giving ( N0 / neNeq ) = 1, from which it follows that

( Neq / Na ) = q = 10p + 1 ………. (Ce)

This expression enables us to place an upper limit to Tm, It has been shown in Appendix B that a galaxy would grow without limit if the total number of origins within its domain always exceeded the rate of extinctions there. For the size to stabilize there must be periods of time during which the rates are reversed and during these periods the average density in the domain must exceed the equilibrium value Neq.

What causes the density to rise above Neq is that the domains of the new clouds that are growing around the older galaxy spread and encroach on the domain of the older galaxy. Thereby the latter loses those regions in which the density is lowest, while it retains the central region in which the density is high. While the mass within the domain remains almost unaltered, the volume of the domain is reduced.

If the boundary of the domain of the new clouds were to advance from all sides up to a mid position between the centres of the new clouds and the older galaxy, the domain of this would be reduced to about one-eighth of its original volume. But the new clouds are much less massive and so the encroachment cannot go so far.

The reduction of the volume of the older galaxy cannot, of course, be sufficient to leave only one-eighth of the previous volume. At the same time the reduction must be great enough to raise the average mass density from less than Na to more than Neq , for if it were not so, if the density in the reduced volume still remained below that given by Neq, the rate of origins within the domain would always exceed the rate of extinctions and the older galaxy would continue without interruption to become ever more massive.

We thus have to conclude, firstly, that fluctuations in the domain of any galaxy range over less than one-eighth in volume, and secondly, that the ratio Neq / Na = q is less than the ratio of the maximum to the minimum volume of the domain.

It would be a complicated matter to calculate the ratio of maximum to minimum volume that should be expected from the growth of new clouds around an existing galaxy, though this will have to be done some day. Meanwhile, Table II will suffice to show how q and Tm are interconnected by equation (Ce). In preparing the table it has been assumed that Ts is 3.66 x 109 years. As p is the ratio of Tm to Ts the half-life of matter Tm is obtained from equation (Cc).

Table II

q

Tm years

5

1.46 x 109

4

1.10 x 109

3

0.73 x 109

2

0.37 x 109

Table I makes it appear unlikely that Tm could be much less than 3 x 108 years. It is unlikely that q can be much greater than 2 and so Table II makes it appear unlikely that Tm can be much greater than 4 x 108 years. If various lines of approach do not allow much room for manoeuvre, they do not at least lead to contradictory conclusions.

It will be shown in Appendix E that a different approach also suggests a value of the order of 4 x 108 years. But if the age of radio-active substances is found to be much greater than 2.6 x 109 years there may be a reason for assigning a longer half-life to matter and, on the other hand, evolutionists may be influenced by the conclusions reached in Appendix E to press for a shorter estimate.

The conclusion that the half-life of matter is shorter than that of some radio-active substances may cause surprise, perhaps be even thought to be impossible. The reason why this conclusion need not be inconsistent with observed facts can only be given after the relation between space and mass has been explored and will be found at the end of Appendix H.

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