Dirac large numbers hypothesis

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:

${\displaystyle \frac{ct}{r_{\text{e}}}\approx 3.47\cdot {10}^{41}\approx {10}^{42},}$

where t is the age of the universe, ${\displaystyle c}$ is the speed of light and r_e is the classical electron radius. Hence, in units where c = 1 and r_e = 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:

${\displaystyle \frac{e^2}{4\pi {ϵ}_{0}G{m}_{\text{p}}{m}_{\text{e}}}\approx {10}^{40}.}$

Hence, interpreting the charge ${\displaystyle e}$ of the electron, the masses ${\displaystyle m_{\text{p}}}$ and ${\displaystyle m_{\text{e}}}$ of the proton and electron, and the permittivity factor ${\displaystyle 4\pi {ϵ}_0}$ in atomic units (equal to 1), the value of the gravitational constant is approximately 10⁻⁴⁰. Dirac interpreted this to mean that ${\displaystyle G}$ varies with time as ${\displaystyle G\approx 1/t}$. George Gamow noted that such a temporal variation does not necessarily follow from Dirac's assumptions, and a corresponding change of G has not been found. According to general relativity, 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:

• '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, with it sparking off many speculations, arguments and new ideas in terms of applications. In the context of geophysics, for instance, Edward Teller seemed to raise a serious objection to LNH in 1948 when he argued that variations in the strength of gravity are not consistent with paleontological data. However, George Gamow demonstrated in 1962 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 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 argued that LNH is inconsistent with experimental results for microwave background radiation whereas Canuto and Hsieh 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) noted that the mass ratio for a typical star (specifically, a star of the Chandrasekhar mass, itself a constant of nature, approx. 1.44 solar masses) and an electron approximates to 10⁶⁰, an interesting variation on the 10⁴⁰ and 10⁸⁰ that are typically associated with Dirac and Eddington respectively. (The physics defining the Chandrasekhar mass produces a ratio that is the −3/2 power of the gravitational fine-structure constant (analogous to the electromagnetic fine-structure constant), 10⁻⁴⁰.)

Modern studies

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) and Matthews (1997) 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. Valev (2019) found an equation connecting cosmological parameters (for example density of the universe) and Planck units (for example Planck density). This ratio of densities, and other ratios (using four fundamental constants: speed of light in vacuum c, Newtonian constant of gravity G, reduced Planck constant ℏ, and Hubble constant H) computes to an exact number, 32.8·10¹²⁰. This provides evidence of the Dirac large numbers hypothesis by connecting the macro-world and the micro-world.