Dirac large numbers hypothesis

The Dirac large numbers hypothesis refers to an observation made by Paul Dirac in 1937 relating ratios of size scales in the Universe to that of force scales. Dirac noted that the ratio of the size of the visible universe, $ct$ with $c$ the speed of light and $t$ the age of the Universe, to the size of a quantum particle $r$ is about $ct/r=10^{40}$ . Hence, in units $c=1$ and $r=1$ , this large number can be taken as the age of the Universe, $t=10^{40}$ .

There is another ratio with this order of magnitude: the ratio of the electrical to the gravitational forces between a proton and an electron, ${\frac {4\pi \epsilon _{0}Gm_{pr}m_{e}}{e^{2}}}$ (≈ 4.4 × 10⁻⁴⁰). Hence, taking the charge $e$ of the electron, the mass $m_{pr}$ / $m_{e}$ of the proton/electron, and the permittivity factor $4\pi \epsilon _{0}$ as units, the gravitational constant equals $G=10^{-40}$ . Dirac interpreted this to mean that $G$ varies with time as $G\propto 1/t\,$ , and built a cosmology out of this idea. If correct, the connection between gravity and quantum mechanics would be unmistakable and may point in the direction of a theory of quantum gravity.

Some scientists believe that the hypothesis is the result of a numerological coincidence, and in 1961, Robert Dicke argued that carbon-based life can only arise when the hypothesis is true lest fusion of hydrogen in stars not occur. A few proponents of non-standard cosmologies refer to Dirac's cosmology as a foundational basis for their ideas.

The Large-Number Coincidence concerning pure numbers of order near $10^{40}$ , was solved in 2006 by Scott Funkhouser (Funkhouser 2006). He found that the coincidence was, as predicted by Weyl, Eddington and Dirac, the result of implicit physical connections, thus validating the Large Numbers Hypothesis. Funkhouser's resolution obviates the need for Dirac's suggestion concerning time-variation of $G$ . In 2008 Funkhouser introduced and resolved a new large-number coincidence involving pure numbers of order near $10^{121}$ . If the two sets of large numbers are related, as expected from the Large Numbers Hypothesis, then the two coincidences would explain the value of the putative cosmological constant (Funkhouser 2008).

References

Dirac, P. A. M. "A New Basis for Cosmology." Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 165, Issue 921, pp. 199-208 (received December 29 1937).
Dirac, P. A. M. "The Cosmological Constants." Nature 139 (1937) 323.
Dirac, P. A. M. "Cosmological Models and the Large Numbers Hypothesis." Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 338, No. 1615, (Jul. 16, 1974), pp. 439-446.
Funkhouser, S. "The Large Number Coincidence, the Cosmic Coincidence and the Critical Acceleration." Proceedings of the Royal Society A 462, 2076 (2006)
Funkhouser, S. "A New Large Number Coincidence and a Scaling Law for the Cosmological Constant." Proceedings of the Royal Society A (2008) 464, 1345-1353

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