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The evaluation of the cosmical number

Published online by Cambridge University Press:  24 October 2008

A. S. Eddington
A. S. Eddington
Affiliation:
The Observatory, Cambridge
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Extract

1. The cosmical number N = . 136 . 2256 is most picturesquely described as ‘the number of protons and electrons in the universe’. This in itself would be a matter of idle curiosity. But N has a more general significance as a fundamental constant which enters into many physical formulae; it determines the ratio of the electrical to the gravitational forces between particles, the range and magnitude of the non-Coulombian forces in atomic nuclei, and the cosmical repulsion manifested in the recession of the nebulae. Its special interpretation as the number of particles in the universe arises in the following way. If we consider a distribution of hydrogen in equilibrium at zero temperature, the presence of the matter produces a curvature of space, and the curvature causes the space to close when the number of particles contained in it reaches a certain total; this total is N. We cannot say with the same confidence that the number of particles in the actual universe is precisely N, because the admission of radiation, complex nuclei, and unsteady conditions takes the problem outside the range of rigorously developed theory; but to the best of our belief these complications do not affect the total number of protons and electrons composing the matter of the universe.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 40 , Issue 1 , March 1944 , pp. 37 - 56
Copyright
Copyright © Cambridge Philosophical Society 1944

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References

* References to the author's other publications are: P. & E., Relativity theory of protons and electrons (Cambridge, 1936)Google Scholar; P.P.S., The philosophy of physical science (Cambridge, 1939); D., The combination of relativity theory and quantum theory, a memoir published by the Dublin Institute of Advanced Studies, Communications, Series A, No. 2. D. contains the determination of all the fundamental physical constants except N; but it does not deal with the E- and EF- frames (wave tensor calculus) of which there is no later account than that given in P. & E.Google Scholar

* The most common kind of combination is ‘averaging’.

* The difficulty of distinguishing measures is less obvious than the difficulty of distinguishing measurables. In a scalar measure there is no difficulty, because the measure-number is the distinction. But a tensor measure is an array of measure-numbers; and we have to define the distinction between different permutations of the numbers in the array, which are, of course, entirely different measures.

* The first round of a structural cycle is always, so to speak, a trial run. In later rounds all quantities are defined cyclically in terms of one another, so that the cycle can be continued forwards or backwards indefinitely.

* In the classification of rank here adopted, two antisymmetrical suffixes count as one; so that a 6-vector (like a 4-vector) is of the first rank, and the Riemann-Christoffel tensor (like the energy tensor) is of the second rank.

* The corresponding theorem in matrix algebra is that any matrix of which the characteristic roots are distinct can be transformed into a diagonal matrix. The limitation to distinct eigenvalues can be disregarded in the physical application, since exact coalescence is a limiting conception which could be approached but never actually reached in natural conditions.

* This does not follow directly from (12), because in (12) as originally derived the sign ambiguities are not all independent, and only 8 combinations are admissible. The 16-fold ambiguity arises at an earlier stage, as explained above. In the simple E-frame a complication occurs, because the reality conditions are such that the real planes of the measure correspond to imaginary planes of the measurable, and vice versa. But since this complication does not occur in the corresponding analysis of the EF-frame (to which the analysis of the E-frame is merely a preliminary) it does not really concern us—except as an indication that the simple frame is inadequate for representing true measurables.

† This refers to the corresponding particles in the double frame.

* This means in effect that we treat a universe composed of hydrogen (as remarked in § 1).

* The analysis in §§ 4–8 was not limited to standard (as opposed to casual) measures; but it is evident that the useful physical application is to standard measures. For there would be no interest in enumerating the total number of casual measurables occupying a given measure.

† We pass straight from the treatment (1) to the treatment (3) in § 10, the intermediate approximation (2) being foreign to our present methods. Thus we do not contemplate a probability distribution over eigenscales, which would be the common way of describing a fluctuation.

* The negative sign attached to dj/j′ 0, which is important in some developments, does not affect the present calculation. It involves a change of K into −K, and therefore a merely nominal change of the measurable from e ikθ to e −ikθ.