Dirac large numbers hypothesis

Last updated November 27, 2023

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 similarity 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: $M\propto t^{2}$ .
Physical constants are actually not constant. Their values depend on the age of the Universe.

Background

LNH was Dirac's personal response to a set of large number "coincidences" that had intrigued other theorists of his time. The "coincidences" began with Hermann Weyl (1919),^[1]^[2] who speculated that the observed radius of the universe, R_U, might also be the hypothetical radius of a particle whose rest energy is equal to the gravitational self-energy of the electron:

{\frac {R_{\text{U}}}{r_{\text{e}}}}\approx {\frac {r_{\text{H}}}{r_{\text{e}}}}\approx 4.1666763\cdot 10^{42}\approx 10^{42.62\ldots },

where,

r_{\text{e}}={\frac {e^{2}}{4\pi \epsilon _{0}\ m_{\text{e}}c^{2}}}\approx 3.7612682\cdot 10^{-16}\mathrm {m}

r_{\text{H}}={\frac {e^{2}}{4\pi \epsilon _{0}\ m_{\text{H}}c^{2}}}\approx 1.5671987\cdot 10^{27}\,\mathrm {m}

with

m_{\text{H}}c^{2}={\frac {Gm_{\text{e}}^{2}}{r_{\text{e}}}}

and r_e is the classical electron radius, m_e is the mass of the electron, m_H denotes the mass of the hypothetical particle, and r_H is its electrostatic radius.

The coincidence was further developed by Arthur Eddington (1931)^[3] who related the above ratios to N, the estimated number of charged particles in the universe^{[ clarification needed ]}:

{\frac {e^{2}}{4\pi \epsilon _{0}\ Gm_{\text{e}}^{2}}}\approx 4.1666763\cdot 10^{42}\approx {\sqrt {N}}

.

In addition to the examples of Weyl and Eddington, Dirac was also influenced by the primeval-atom hypothesis of Georges Lemaître, who lectured on the topic in Cambridge in 1933. 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.^[4]^[5] For Milne, space was not a structured object but simply a system of reference in which relations such as this could accommodate Einstein's conclusions:

G=\left(\!{\frac {c^{3}}{M_{\text{U}}}}\!\right)t,

where M_U is the mass of the universe and t is the age of the universe. 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 {c\,t}{r_{\text{e}}}}\approx 3.47\cdot 10^{41}\approx 10^{42},

where t is the age of the universe, $c$ 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:

{\frac {e^{2}}{4\pi \epsilon _{0}Gm_{\text{p}}m_{\text{e}}}}\approx 10^{40}.

Hence, interpreting the charge $e$ of the electron, the masses $m_{\text{p}}$ and $m_{\text{e}}$ of the proton and 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 Gamow noted that such a temporal variation does not necessarily follow from Dirac's assumptions,^[6] a corresponding change of G has not been found.^[7] 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:

'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.^[8] In the context of geophysics, for instance, Edward Teller seemed to raise a serious objection to LNH in 1948^[9] when he argued that variations in the strength of gravity are not consistent with paleontological data. However, George Gamow demonstrated in 1962^[10] 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^[11] 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^[12] argued that LNH is inconsistent with experimental results for microwave background radiation whereas Canuto and Hsieh^[13]^[14] 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)^[15] 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, 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)^[16] and Matthews (1997)^[17] 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.^[18] Valev (2019)^[19] 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.

Related Research Articles

<span class="mw-page-title-main">Hubble's law</span> Observation in physical cosmology

Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving away from Earth. The velocity of the galaxies has been determined by their redshift, a shift of the light they emit toward the red end of the visible spectrum.

In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by $α$ , is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between elementary charged particles.

The Hartree atomic units are a system of natural units of measurement which is especially convenient for calculations in atomic physics and related scientific fields, such as computational chemistry and atomic spectroscopy. They are named after the physicist Douglas Hartree. Atomic units are often abbreviated "a.u." or "au", not to be confused with the same abbreviation used also for astronomical units, arbitrary units, and absorbance units in other contexts.

The Eddington luminosity, also referred to as the Eddington limit, is the maximum luminosity a body can achieve when there is balance between the force of radiation acting outward and the gravitational force acting inward. The state of balance is called hydrostatic equilibrium. When a star exceeds the Eddington luminosity, it will initiate a very intense radiation-driven stellar wind from its outer layers. Since most massive stars have luminosities far below the Eddington luminosity, their winds are mostly driven by the less intense line absorption. The Eddington limit is invoked to explain the observed luminosity of accreting black holes such as quasars.

In physics, a dimensionless physical constant is a physical constant that is dimensionless, i.e. a pure number having no units attached and having a numerical value that is independent of whatever system of units may be used.

In astrophysics, the Eddington number, $N Edd$ , is the number of protons in the observable universe. Eddington originally calculated it as about 1.57×10⁷⁹; current estimates make it approximately 10⁸⁰.

The horizon problem is a cosmological fine-tuning problem within the Big Bang model of the universe. It arises due to the difficulty in explaining the observed homogeneity of causally disconnected regions of space in the absence of a mechanism that sets the same initial conditions everywhere. It was first pointed out by Wolfgang Rindler in 1956.

In particle physics, the Klein–Nishina formula gives the differential cross section of photons scattered from a single free electron, calculated in the lowest order of quantum electrodynamics. It was first derived in 1928 by Oskar Klein and Yoshio Nishina, constituting one of the first successful applications of the Dirac equation. The formula describes both the Thomson scattering of low energy photons and the Compton scattering of high energy photons, showing that the total cross section and expected deflection angle decrease with increasing photon energy.

Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to calculate elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example is an explanation for the color of gold: due to relativistic effects, it is not silvery like most other metals.

A variable speed of light (VSL) is a feature of a family of hypotheses stating that the speed of light may in some way not be constant, for example, that it varies in space or time, or depending on frequency. Accepted classical theories of physics, and in particular general relativity, predict a constant speed of light in any local frame of reference and in some situations these predict apparent variations of the speed of light depending on frame of reference, but this article does not refer to this as a variable speed of light. Various alternative theories of gravitation and cosmology, many of them non-mainstream, incorporate variations in the local speed of light.

In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin and electric charge. The value of the electron magnetic moment is −9.2847647043(28)×10⁻²⁴ J⋅T⁻¹. In units of the Bohr magneton (μ_B), it is −1.00115965218059(13) μ_B, a value that was measured with a relative accuracy of 1.3×10⁻¹³.

In physics, there is a speculative hypothesis that, if there were a black hole with the same mass, charge and angular momentum as an electron, it would share other properties of the electron. Most notably, Brandon Carter showed in 1968 that the magnetic moment of such an object would match that of an electron. This is interesting because calculations ignoring special relativity and treating the electron as a small rotating sphere of charge give a magnetic moment roughly half the experimental value.

The cosmic neutrino background is the universe's background particle radiation composed of neutrinos. They are sometimes known as relic neutrinos.

The Einstein–de Haas effect is a physical phenomenon in which a change in the magnetic moment of a free body causes this body to rotate. The effect is a consequence of the conservation of angular momentum. It is strong enough to be observable in ferromagnetic materials. The experimental observation and accurate measurement of the effect demonstrated that the phenomenon of magnetization is caused by the alignment (polarization) of the angular momenta of the electrons in the material along the axis of magnetization. These measurements also allow the separation of the two contributions to the magnetization: that which is associated with the spin and with the orbital motion of the electrons. The effect also demonstrated the close relation between the notions of angular momentum in classical and in quantum physics.

In cosmology, baryon acoustic oscillations (BAO) are fluctuations in the density of the visible baryonic matter of the universe, caused by acoustic density waves in the primordial plasma of the early universe. In the same way that supernovae provide a "standard candle" for astronomical observations, BAO matter clustering provides a "standard ruler" for length scale in cosmology. The length of this standard ruler is given by the maximum distance the acoustic waves could travel in the primordial plasma before the plasma cooled to the point where it became neutral atoms, which stopped the expansion of the plasma density waves, "freezing" them into place. The length of this standard ruler can be measured by looking at the large scale structure of matter using astronomical surveys. BAO measurements help cosmologists understand more about the nature of dark energy by constraining cosmological parameters.

In physics the Stoney units form a system of units named after the Irish physicist George Johnstone Stoney, who first proposed them in 1881. They are the earliest example of natural units, i.e., a coherent set of units of measurement designed so that chosen physical constants fully define and are included in the set.

In cosmology, recombination refers to the epoch during which charged electrons and protons first became bound to form electrically neutral hydrogen atoms. Recombination occurred about 378,000 years after the Big Bang. The word "recombination" is misleading, since the Big Bang theory doesn't posit that protons and electrons had been combined before, but the name exists for historical reasons since it was named before the Big Bang hypothesis became the primary theory of the birth of the universe.

In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: c, G, ħ, and k_B. Expressing one of these physical constants in terms of Planck units yields a numerical value of 1. They are a system of natural units, defined using fundamental properties of nature rather than properties of a chosen prototype object. Originally proposed in 1899 by German physicist Max Planck, they are relevant in research on unified theories such as quantum gravity.

Electric dipole spin resonance (EDSR) is a method to control the magnetic moments inside a material using quantum mechanical effects like the spin–orbit interaction. Mainly, EDSR allows to flip the orientation of the magnetic moments through the use of electromagnetic radiation at resonant frequencies. EDSR was first proposed by Emmanuel Rashba.

The term physical constant expresses the notion of a physical quantity subject to experimental measurement which is independent of the time or location of the experiment. The constancy (immutability) of any "physical constant" is thus subject to experimental verification.

References

↑ H. Weyl (1917). "Zur Gravitationstheorie". Annalen der Physik (in German). 359 (18): 117–145. Bibcode:1917AnP...359..117W. doi:10.1002/andp.19173591804.
↑ H. Weyl (1919). "Eine neue Erweiterung der Relativitätstheorie". Annalen der Physik . 364 (10): 101–133. Bibcode:1919AnP...364..101W. doi:10.1002/andp.19193641002.
↑ A. Eddington (1931). "Preliminary Note on the Masses of the Electron, the Proton, and the Universe". Proceedings of the Cambridge Philosophical Society . 27 (1): 15–19. Bibcode:1931PCPS...27...15E. doi:10.1017/S0305004100009269. S2CID 122865789.
↑ E. A. Milne (1935). Relativity, Gravity and World Structure. Oxford University Press.
↑ H. Kragh (1996). Cosmology and Controversy: The historical development of two theories of the universe. Princeton University Press. pp. 61–62. ISBN 978-0-691-02623-7.
↑ H. Kragh (1990). Dirac: A Scientific Biography . Cambridge University Press. p. 177. ISBN 978-0-521-38089-8.
↑ J. P.Uzan (2003). "The fundamental constants and their variation, Observational status and theoretical motivations". Reviews of Modern Physics . 75 (2): 403. arXiv: hep-ph/0205340 . Bibcode:2003RvMP...75..403U. doi:10.1103/RevModPhys.75.403. S2CID 118684485.
↑ Saibal, Ray; Mukhopadhyay, Utpal; Ray, Soham; Bhattacharjee, Arjak (2019). "Dirac's large number hypothesis: A journey from concept to implication". International Journal of Modern Physics D. 28 (8) – via World Scientific.
↑ E. Teller (1948). "On the change of physical constants". Physical Review . 73 (7): 801–802. Bibcode:1948PhRv...73..801T. doi:10.1103/PhysRev.73.801.
↑ G. Gamow (1962). Gravity. Doubleday. pp. 138–141. LCCN 62008840.
↑ G. Blake (1978). "The Large Numbers Hypothesis and the rotation of the Earth". Monthly Notices of the Royal Astronomical Society . 185 (2): 399–408. Bibcode:1978MNRAS.185..399B. doi: 10.1093/mnras/185.2.399 .
↑ D. Falik (1979). "Primordial Nucleosynthesis and Dirac's Large Numbers Hypothesis". The Astrophysical Journal . 231: L1. Bibcode:1979ApJ...231L...1F. doi:10.1086/182993.
↑ V. Canuto, S. Hsieh (1978). "The 3 K blackbody radiation, Dirac's Large Numbers Hypothesis, and scale-covariant cosmology". The Astrophysical Journal . 224: 302. Bibcode:1978ApJ...224..302C. doi:10.1086/156378.
↑ V. Canuto, S. Hsieh (1980). "Primordial nucleosynthesis and Dirac's large numbers hypothesis". The Astrophysical Journal . 239: L91. Bibcode:1980ApJ...239L..91C. doi: 10.1086/183299 .
↑ P. Jordan (1947). "Die Herkunft der Sterne". Astronomische Nachrichten. 275 (10–12): 191. Bibcode:1947dhds.book.....J. doi:10.1002/asna.19472751012.
↑ L. Nottale. "Mach's Principle, Dirac's Large Numbers and the Cosmological Constant Problem" (PDF).
↑ R. Matthews (1998). "Dirac's coincidences sixty years on". Astronomy & Geophysics . 39 (6): 19–20. doi: 10.1093/astrog/39.6.6.19 .
↑ H. Lyre (2003). "C. F. Weizsäcker's Reconstruction of Physics: Yesterday, Today and Tomorrow". arXiv: quant-ph/0309183 .
↑ D. Valev (2019). "Evidence of Dirac large numbers hypothesis" (PDF). Proceedings of the Romanian Academy. 20 (+4): 361–368.

External links

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] H. Weyl (1917). "Zur Gravitationstheorie". Annalen der Physik (in German). 359 (18): 117–145. Bibcode:1917AnP...359..117W. doi:10.1002/andp.19173591804.

[2] H. Weyl (1919). "Eine neue Erweiterung der Relativitätstheorie". Annalen der Physik . 364 (10): 101–133. Bibcode:1919AnP...364..101W. doi:10.1002/andp.19193641002.

[3] A. Eddington (1931). "Preliminary Note on the Masses of the Electron, the Proton, and the Universe". Proceedings of the Cambridge Philosophical Society . 27 (1): 15–19. Bibcode:1931PCPS...27...15E. doi:10.1017/S0305004100009269. S2CID 122865789.

[4] E. A. Milne (1935). Relativity, Gravity and World Structure. Oxford University Press.

[5] H. Kragh (1996). Cosmology and Controversy: The historical development of two theories of the universe. Princeton University Press. pp. 61–62. ISBN 978-0-691-02623-7.

[6] H. Kragh (1990). Dirac: A Scientific Biography . Cambridge University Press. p. 177. ISBN 978-0-521-38089-8.

[7] J. P.Uzan (2003). "The fundamental constants and their variation, Observational status and theoretical motivations". Reviews of Modern Physics . 75 (2): 403. arXiv: hep-ph/0205340 . Bibcode:2003RvMP...75..403U. doi:10.1103/RevModPhys.75.403. S2CID 118684485.

[8] Saibal, Ray; Mukhopadhyay, Utpal; Ray, Soham; Bhattacharjee, Arjak (2019). "Dirac's large number hypothesis: A journey from concept to implication". International Journal of Modern Physics D. 28 (8) – via World Scientific.

[9] E. Teller (1948). "On the change of physical constants". Physical Review . 73 (7): 801–802. Bibcode:1948PhRv...73..801T. doi:10.1103/PhysRev.73.801.

[10] G. Gamow (1962). Gravity. Doubleday. pp. 138–141. LCCN 62008840.

[11] G. Blake (1978). "The Large Numbers Hypothesis and the rotation of the Earth". Monthly Notices of the Royal Astronomical Society . 185 (2): 399–408. Bibcode:1978MNRAS.185..399B. doi: 10.1093/mnras/185.2.399 .

[12] D. Falik (1979). "Primordial Nucleosynthesis and Dirac's Large Numbers Hypothesis". The Astrophysical Journal . 231: L1. Bibcode:1979ApJ...231L...1F. doi:10.1086/182993.

[13] V. Canuto, S. Hsieh (1978). "The 3 K blackbody radiation, Dirac's Large Numbers Hypothesis, and scale-covariant cosmology". The Astrophysical Journal . 224: 302. Bibcode:1978ApJ...224..302C. doi:10.1086/156378.

[14] V. Canuto, S. Hsieh (1980). "Primordial nucleosynthesis and Dirac's large numbers hypothesis". The Astrophysical Journal . 239: L91. Bibcode:1980ApJ...239L..91C. doi: 10.1086/183299 .

[15] P. Jordan (1947). "Die Herkunft der Sterne". Astronomische Nachrichten. 275 (10–12): 191. Bibcode:1947dhds.book.....J. doi:10.1002/asna.19472751012.

[16] L. Nottale. "Mach's Principle, Dirac's Large Numbers and the Cosmological Constant Problem" (PDF).

[17] R. Matthews (1998). "Dirac's coincidences sixty years on". Astronomy & Geophysics . 39 (6): 19–20. doi: 10.1093/astrog/39.6.6.19 .

[18] H. Lyre (2003). "C. F. Weizsäcker's Reconstruction of Physics: Yesterday, Today and Tomorrow". arXiv: quant-ph/0309183 .

[19] D. Valev (2019). "Evidence of Dirac large numbers hypothesis" (PDF). Proceedings of the Romanian Academy. 20 (+4): 361–368.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]