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
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.)
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.)
Number of elementary components that originate in unit volume
Fraction of elementary components in any volume that become
extinct in unit time.
Time during which the linear dimensions of space double.
Half-life of matter.
Tm / Ts
Neq / Na
Mass of the earth at any time t
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
This can be expressed by the equation
dm / dt = -ne m
m = m0e net
m0 = menet
ln(m0 / m) = net
For the half-life of matter
t= Tm and ( m0 / m ) = 2
ln2 = 0.69 = neTm
It follows that for any values of m0 and t
ln( m0 / m )= 0.69t / Tm
For small values of t/Tm this can be replaced by
(m0 / m) = 1 + 0.69 t / Tm
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:
Tm x 109 years
m0 / m
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
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
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
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
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
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).
1.46 x 109
1.10 x 109
0.73 x 109
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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