# Dirac large numbers hypothesis

Paul Dirac

The Dirac large numbers hypothesis (LNH) is an observation made by Paul Dirac in 1937 relating ratios of size scales in the Universe to that of force scales. The ratios constitute very large, dimensionless numbers: some 40 orders of magnitude in the present cosmological epoch. According to Dirac's hypothesis, the apparent equivalence of these ratios might not be a mere coincidence but instead could imply a cosmology with these unusual features:

• The strength of gravity, as represented by the gravitational constant, is inversely proportional to the age of the universe: $G \propto 1/t\,$
• The mass of the universe is proportional to the square of the universe's age: $M \propto t^2$.

Neither of these two features has gained wide acceptance in mainstream physics and, though some proponents[who?] of non-standard cosmologies refer to Dirac's cosmology as a foundational basis for their own ideas and studies, some physicists[who?] dismiss the large numbers in LNH as mere coincidences. A coincidence, however, may be defined optimally as 'an event that provides support for an alternative to a currently favoured causal theory, but not necessarily enough support to accept that alternative in light of its low prior probability.'[1] Research into LNH, or the large number of coincidences that underpin it, appears to have gained new impetus from failures in standard cosmology to account for anomalies such as the recent discovery that the universe might be expanding at an accelerated rate.[2]

## Background

LNH was Dirac's personal response to a set of large number 'coincidences' that had intrigued other theorists at about the same time. The 'coincidences' began with Hermann Weyl (1919),[2][3][4] who speculated that the observed radius of the universe, RU, might also be the hypothetical radius of a particle whose rest energy is equal to the gravitational self-energy of the electron:

$\frac {R_U}{r_e}\approx \frac{r_H}{r_e} \approx 10^{42} ,$
$r_e = \frac {e^2}{4 \pi \epsilon_0 m_e c^2},$
$r_H = \frac {e^2}{4 \pi \epsilon_0 m_H c^2},$
$m_H c^2 = \frac {Gm_e^2}{r_e}$

where re is the classical electron radius, me is the mass of the electron, mH denotes the mass of the hypothetical particle, and rH is its electrostatic radius.

The coincidence was further developed by Arthur Eddington (1931) [5] who related the above ratios to N, the estimated number of charged particles in the Universe:

$\frac {e^2}{4 \pi \epsilon_0 Gm_e^2} \approx \sqrt {N} \approx 10^{42}.$

In addition to the examples of Weyl and Eddington, Dirac was influenced also by the primeval-atom hypothesis of Georges Lemaître, who lectured on the topic in Cambridge in 1933.[2] The notion of a varying-G cosmology first appears in the work of Edward Arthur Milne a few years before Dirac formulated LNH. Milne was inspired not by large number coincidences but by a dislike of Einstein's general theory of relativity.[6][7] For Milne, space was not a structured object but simply a system of reference in which Einstein's conclusions could be accommodated by relations such as this:

$G = \left(\frac{c^3}{M_U}\right)t,$

where MU is the mass of the universe and t is the age of the universe in seconds. According to this relation, G increases over time.

## Dirac's interpretation of the large number coincidences

The Weyl and Eddington ratios above can be rephrased in a variety of ways, as for instance in the context of time:

$\frac {ct}{r_e} \approx 10^{40},$

where t is the age of the universe, $c$ is the speed of light and re is the classical electron radius. Hence, in units where c=1 and re = 1, the age of the Universe is about 1040 units of time. This is the same order of magnitude as the ratio of the electrical to the gravitational forces between a proton and an electron:

$\frac{4 \pi \epsilon_0 G m_p m_e}{e^2} \approx 10^{-40}.$

Hence, interpreting the charge $e$ of the electron, the mass $m_p$/$m_e$ of the proton/electron, and the permittivity factor $4 \pi \epsilon_0$ in atomic units (equal to 1), the value of the gravitational constant is approximately 10−40. Dirac interpreted this to mean that $G$ varies with time as $G \approx 1/t\,$. Although George Gamov noted that such a temporal variation does not necessarily follow from Dirac's assumptions,[8] a corresponding change of G has not been found.[9] According to general relativity, however, G is constant, otherwise the law of conserved energy is violated. Dirac met this difficulty by introducing into the Einstein field equations a gauge function β that describes the structure of spacetime in terms of a ratio of gravitational and electromagnetic units. He also provided alternative scenarios for the continuous creation of matter, one of the other significant issues in LNH:[2]

• 'additive' creation (new matter is created uniformly throughout space) and
• 'multiplicative' creation (new matter is created where there are already concentrations of mass).

## Later developments and interpretations

Dirac's theory has inspired and continues to inspire a significant body of scientific literature in a variety of disciplines. In the context of geophysics, for instance, Edward Teller seemed to raise a serious objection to LNH in 1948 [10] when he argued that variations in the strength of gravity are not consistent with paleontological data. However, George Gamow demonstrated in 1962 [11] how a simple revision of the parameters (in this case, the age of the solar system) can invalidate Teller's conclusions. The debate is further complicated by the choice of LNH cosmologies: In 1978, G. Blake [12] argued that paleontological data is consistent with the 'multiplicative' scenario but not the 'additive' scenario. Arguments both for and against LNH are also made from astrophysical considerations. For example, D. Falik[13] argued that LNH is inconsistent with experimental results for microwave background radiation whereas Canuto and Hsieh[14][15] argued that it is consistent. One argument that has created significant controversy was put forward by Robert Dicke in 1961. Known as the anthropic coincidence or fine-tuned universe, it simply states that the large numbers in LNH are a necessary coincidence for intelligent beings since they parametrize fusion of hydrogen in stars and hence carbon-based life would not arise otherwise.

Various authors have introduced new sets of numbers into the original 'coincidence' considered by Dirac and his contemporaries, thus broadening or even departing from Dirac's own conclusions. Jordan (1947) [16] noted that the mass ratio for a typical star and an electron approximates to 1060, an interesting variation on the 1040 and 1080 that are typically associated with Dirac and Eddington respectively. Various numbers of the order of 1060 were arrived at by V. E. Shemi-Zadah (2002) [17] through measuring cosmological entities in Planck units. P. Zizzi (1998) argued that there might be a modern mathematical interpretation of LNH in a Planck-scale setting in the context of quantum foam.[18] The relevance of the Planck scale to LNH was further demonstrated by S. Caneiro and G. Marugan (2002)[19] by reference to the holographic principle. Previously, Carneiro (1997)[20] arrived at an intermediate scaling factor 1020 when considering the possible quantization of cosmic structures and a rescaling of Planck's constant.

Several authors have recently identified and pondered the significance of yet another large number, approximately 120 orders of magnitude. This is for example the ratio of the theoretical and observational estimates of the energy density of the vacuum, which Nottale (1993)[21] and Matthews (1997)[22] associated in an LNH context with a scaling law for the cosmological constant. Carl Friedrich von Weizsäcker identified 10120 with the ratio of the universe's volume to the volume of a typical nucleon bounded by its Compton wavelength, and he identified this ratio with the sum of elementary events or bits of information in the universe.[23] T. Goernitz (1986), building on Weizsäcker's work, posited an explanation for large number 'coincidences' in the context of Bekenstein–Hawking entropy.[24] Genreith (1999)[25] has sketched out a fractal cosmology in which the smallest mass, which he identified as a neutrino, is about 120 orders of magnitude smaller than the mass of the universe (note: this 'neutrino' approximates in scale to the hypothetical particle mH mentioned above in the context of Weyl's work in 1919). Sidharth (2005)[26] interpreted a typical electromagnetic particle such as the pion as a collection of 1040 Planck oscillators and the universe as a collection of 10120 Planck oscillators. The fact that a number like 10120 can be represented in a variety of ways has been interpreted by Funkhouser (2006)[27] as a new large numbers coincidence. Funkhouser claimed to have 'resolved' the LNH coincidences without departing from the standard model for cosmology. In a similar vein, Carneiro and Marugan (2002) claimed that the scaling relations in LNH can be explained entirely according to basic principles.[19]

## References

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