Appendix to Newton's Gravitation Constant G as a Quantum Coupling Constant

by

James G. Gilson

Queen Mary, University of London

October 3rd 2004

1. Generalized formula for G

This appendix is devoted to an application of the formula

equation (8) in the article Newton's Gravitation Constant, G, as a Quantum Coupling Constant. This is a generalization of the formula suggested by Ross McPherson's¹ who takes ς = 1. Here the quantity ς has the units form of time and is not necessarily equal to unity but is assumed to have a numerical value of the order unity. ν_p = m_p c²/h. That is to say, it is the proton's rest energy quantum frequency. If this is substituted into formula (1), it becomes

The combination ς 10¹⁶/(2 π) will also have the same dimensions form as does ς. That is time so we shall denote it by T = t_n + t_o, the n standing for now. The t_o is a base value it would have if for some reason the part t_n were to assume the value zero. The reason for this addition will be explained as we proceed,

The reliability situation with regard to this number is not good. There is no really certain cosmology theory. It is not even certain that the very concept of an age of the universe is meaningful. Thus t_a can only be used as a very rough guide in comparing theories. Thus the age T would represent a young universe but under the condition of very uncertain knowledge that we have, T = t_n + t_o is a suitable candidate to fit the bill of being the age of the universe. Using this quantity and simultaneously recognising that G is a function of t_n, formula (2) becomes

It is unlikely that any formula of this type will apply at the very early stages of the universe so I now use the base time value t_o to impose the condition that formula (6) only applies for times, t_n > 0, acknowledging that for t_n ≤ 0 conditions in the universe may be so extreme that either the formula for G(t_n) in terms of time will be different or even possibly the concept of gravity may not be meaningful in any present day human terms. Formula (6) has the feature in common with the Hoyle Narlikar cosmological theory⁸ that the age of the universe appears linearly in the denominator of the gravitational constant. This give the clue of how the Hubble constant H should be obtained from the construction here. Following the Hoyle Narlikar prescription, we can define H as,

where a is a constant of order unity. However, here it is clear that a is not just of order unity it is exactly unity. Hence we can write,

This is clearly true if the current age of the universe T = t_n + t_o and the current radius of the universe R is defined as c T.

Equation (9) represents the usual perception of Hubble's law that the radial velocities of distant gallaxies are proportional to their distance from the observer. Thus the whole story of the expansion of the universe is contained in the form of the gravitation constant expressed as

2. Validation of McPherson's formula for G

McPherson's¹observation is that the pure number ratio R_PE of the electromagnetic potential energy between a proton and an electron to the gravitational potential energy between a proton and an electron is very well approximated by the quantum frequency ν_p of the proton multiplied by P=10¹⁶. This led him to the formula (1) in this article with numerically ς = 1 and dimensionless. However, with the usual understanding of the meanings and dimensions of the physical quantities involved the dimensions of G implied in formula (1) would be k_g^-1m³s^-1 whereas the usual physical G does in fact have the dimensions k_g^-1m³s^-2. Thus there is a time dimension missing in the formula unless ς is assumed to have the dimensions form of time. This anomaly is clearly the result of equating quantities that although having a closely similar sequence of digits appearing are of different physical dimensional character. The numerical quantity multiplier 10¹⁶ used to render the two quantities equated numerically almost equal is clutched out of thin air and indeed it seems very unlikely that a power of 10 integer like this would appear in a fundamental physical formula. Thus on the face of it, one could easily conclude that the formula is physically meaningless. However, as shown in the paper for which this is the appendix and here in the appendix, the use of the time dimensioned quantity ς removes all of these possible objections. Here it has been shown that the ς 10¹⁶/(2 π) should be identified with the current age of the universe. Thus in addition to the formula having the power to open the way to an electromagnetic theory of gravity, it also has the power to open the way to a direct and simple theoretical understanding of one of the most significant scientific philosophical advances of the 20th century, Hubble's discovery in 1929 that The Universe is Expanding.

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