Editing Dirac large numbers hypothesis

{{Short description|Hypothesis relating age of the universe to physical constants}}
[[Image:Dirac 4.jpg|thumb|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 [[Orders of magnitude (numbers)#1039|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 [[physical cosmology|cosmology]] with these unusual features:
*The strength of gravity, as represented by the [[gravitational constant]], is inversely proportional to the [[age of the universe]]: <math>G \propto 1/t\,</math>
*The mass of the universe is proportional to the square of the universe's age: <math>M \propto t^2</math>.
*Physical constants are actually not constant. Their values depend on the age of the Universe.
Stated in another way, the hypothesis states that all very large dimensionless quantities occurring in fundamental physics should be simply related to a single very large number, which Dirac chose to be the age of the universe.<ref>Giudice, Gian Francesco. "Naturally speaking: the naturalness criterion and physics at the LHC." ''Perspectives on LHC physics'' (2008): 155-178.</ref>

== 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),<ref>{{cite journal|author=H. Weyl|year=1917|title=Zur Gravitationstheorie|journal=[[Annalen der Physik]]|volume=359|issue=18|pages=117–145|bibcode=1917AnP...359..117W|doi=10.1002/andp.19173591804|url=https://zenodo.org/record/1424330|language=de}}</ref><ref>{{cite journal|author=H. Weyl|year=1919|title=Eine neue Erweiterung der Relativitätstheorie|journal=[[Annalen der Physik]]|volume=364|issue=10|pages=101–133|bibcode=1919AnP...364..101W|doi=10.1002/andp.19193641002|url=https://zenodo.org/record/1424345}}</ref> who speculated that the observed radius of the universe, ''R''<sub>U</sub>, might also be the hypothetical radius of a particle whose rest energy is equal to the gravitational self-energy of the electron:
:<math>\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} ,</math>
where,
:<math>r_\text{e} = \frac {e^2}{4 \pi \epsilon_0 \ m_\text{e} c^2} \approx 3.7612682 \cdot 10^{-16} \mathrm{m}</math>
:<math>r_\text{H} = \frac {e^2}{4 \pi \epsilon_0 \ m_\text{H} c^2} \approx 1.5671987 \cdot 10^{27} 
 \,\mathrm{m}</math> with <math>m_\text{H} c^2 = \frac {Gm_\text{e}^2}{r_\text{e}}</math>
and ''r''<sub>e</sub> is the [[classical electron radius]], ''m''<sub>e</sub> is the mass of the electron, ''m''<sub>H</sub> denotes the mass of the hypothetical particle, and ''r''<sub>H</sub> is its electrostatic radius.

The coincidence was further developed by [[Arthur Eddington]] (1931)<ref>
{{cite journal
 |author=A. Eddington
 |year=1931
 |title=Preliminary Note on the Masses of the Electron, the Proton, and the Universe
 |journal=[[Proceedings of the Cambridge Philosophical Society]]
 |volume=27 |issue= 1|pages=15–19
 |bibcode=1931PCPS...27...15E
 |doi=10.1017/S0305004100009269
|s2cid=122865789
 }}</ref> who related the above ratios to '''N''', the estimated number of charged particles in the universe, with the following ratio:<ref name=":0" />
:<math>\frac {e^2}{4 \pi \epsilon_0 \ Gm_\text{e}^2} \approx 4.1666763 \cdot 10^{42} \approx \sqrt {N}</math>.

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]].<ref>
{{cite book
 |author=E. A. Milne
 |year=1935
 |title=Relativity, Gravity and World Structure
 |publisher=[[Oxford University Press]]
}}</ref><ref>
{{cite book
 |author=H. Kragh
 |year=1996
 |title=Cosmology and Controversy: The historical development of two theories of the universe
 |pages=[https://archive.org/details/cosmologycontrov00helg/page/61 61–62]
 |publisher=[[Princeton University Press]]
 |isbn=978-0-691-02623-7
 |url=https://archive.org/details/cosmologycontrov00helg/page/61
 }}</ref> 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:
:<math>G = \left(\!\frac{c^3}{M_\text{U}}\!\right)t,</math>
where ''M''<sub>U</sub> 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:
:<math>\frac {c\,t}{r_\text{e}} \approx 3.47 \cdot 10^{41} \approx 10^{42},</math>

where ''t'' is the age of the universe, <math>c</math> is the [[speed of light]] and ''r''<sub>e</sub> is the classical electron radius. Hence, in units where {{nowrap|1=''c'' = 1}} and {{nowrap|1=''r''<sub>e</sub> = 1}}, the age of the universe is about 10<sup>40</sup> units of time. This is the same [[order of magnitude]] as the ratio of the [[electromagnetic force|electrical]] to the [[gravitational]] [[force]]s between a [[proton]] and an [[electron]]: 
:<math>\frac{e^2}{4 \pi \epsilon_0 G m_\text{p} m_\text{e}} \approx 10^{40}.</math>

Hence, interpreting the [[electric charge|charge]] <math>e</math> of the [[electron]], the [[mass]]es <math>m_\text{p}</math> and <math>m_\text{e}</math> of the proton and electron, and the permittivity factor <math> 4 \pi \epsilon_0</math> in atomic units (equal to 1), the value of the [[gravitational constant]] is approximately 10<sup>−40</sup>. Dirac interpreted this to mean that <math>G</math> varies with time as <math>G \approx 1/t</math>. Although [[George Gamow]] noted that such a temporal variation does not necessarily follow from Dirac's assumptions,<ref>
{{cite book
 |author=H. Kragh
 |year=1990
 |title=Dirac: A Scientific Biography
 |url=https://archive.org/details/diracscientificb0000krag
 |url-access=registration
 |publisher=[[Cambridge University Press]]
 |page=[https://archive.org/details/diracscientificb0000krag/page/177 177]
 |isbn=978-0-521-38089-8
}}</ref> a corresponding change of ''G'' has not been found.<ref>
{{cite journal
 |author=J. P.Uzan
 |year=2003
 |title=The fundamental constants and their variation, Observational status and theoretical motivations
 |journal=[[Reviews of Modern Physics]]
 |volume=75 |issue=2 |page=403
 |arxiv=hep-ph/0205340
 |bibcode=2003RvMP...75..403U
 |doi=10.1103/RevModPhys.75.403
|s2cid=118684485
 }}</ref>
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 {{lang|grc|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.<ref>{{Cite journal |last1=Saibal |first1=Ray |last2=Mukhopadhyay |first2=Utpal |last3=Ray |first3=Soham |last4=Bhattacharjee |first4=Arjak |date=2019 |title=Dirac's large number hypothesis: A journey from concept to implication |url=https://www.worldscientific.com/doi/10.1142/S0218271819300143 |journal=International Journal of Modern Physics D |volume=28 |issue=8 |pages=1930014–1930096 |doi=10.1142/S0218271819300143 |bibcode=2019IJMPD..2830014R |via=World Scientific|url-access=subscription }}</ref> In the context of [[geophysics]], for instance, [[Edward Teller]] seemed to raise a serious objection to LNH in 1948<ref>
{{cite journal
 |author=E. Teller
 |year=1948
 |title=On the change of physical constants
 |journal=[[Physical Review]]
 |volume=73 |issue=7 |pages=801–802
 |bibcode=1948PhRv...73..801T
 |doi=10.1103/PhysRev.73.801
}}</ref> when he argued that variations in the strength of gravity are not consistent with [[paleontology|paleontological]] data. However, [[George Gamow]] demonstrated in 1962<ref>
{{cite book
 |author=G. Gamow
 |year=1962
 |title=Gravity
 |pages=138–141
 |publisher=[[Doubleday (publisher)|Doubleday]]
 |lccn=62008840
}}</ref> 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 [[Cosmology|cosmologies]]: In 1978, G. Blake<ref>
{{cite journal
 |author=G. Blake
 |year=1978
 |title=The Large Numbers Hypothesis and the rotation of the Earth
 |journal=[[Monthly Notices of the Royal Astronomical Society]]
 |volume=185 |issue=2
 |pages=399–408
 |bibcode=1978MNRAS.185..399B
 |doi=10.1093/mnras/185.2.399
 |doi-access=free
}}</ref> 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<ref>
{{cite journal
 |author=D. Falik
 |year=1979
 |title=Primordial Nucleosynthesis and Dirac's Large Numbers Hypothesis
 |journal=[[The Astrophysical Journal]]
 |volume=231 |page=L1
 |bibcode=1979ApJ...231L...1F
 |doi=10.1086/182993
}}</ref> argued that LNH is inconsistent with experimental results for [[microwave background radiation]] whereas Canuto and Hsieh<ref>
{{cite journal
 |author=V. Canuto, S. Hsieh
 |year=1978
 |title=The 3 K blackbody radiation, Dirac's Large Numbers Hypothesis, and scale-covariant cosmology
 |journal=[[The Astrophysical Journal]]
 |volume=224 |pages=302
 |bibcode=1978ApJ...224..302C
 |doi=10.1086/156378
}}</ref><ref>
{{cite journal
 |author=V. Canuto, S. Hsieh
 |year=1980
 |title=Primordial nucleosynthesis and Dirac's large numbers hypothesis
 |journal=[[The Astrophysical Journal]]
 |volume=239 |pages=L91
 |bibcode=1980ApJ...239L..91C
 |doi=10.1086/183299
|doi-access=free
 }}</ref> 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 [[nuclear fusion|fusion]] of [[hydrogen]] in [[star]]s 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)<ref>
{{Cite journal
 |author=P. Jordan
 |year=1947
 |title=Die Herkunft der Sterne
 |journal=Astronomische Nachrichten
 |volume=275
 |issue=10–12
 |pages=191
 |bibcode=1947dhds.book.....J
|doi=10.1002/asna.19472751012
}}</ref> noted that the mass ratio for a typical star (specifically, a star of the [[Chandrasekhar limit|Chandrasekhar mass]], itself a constant of nature, approx. 1.44 solar masses) and an electron approximates to 10<sup>60</sup>, an interesting variation on the 10<sup>40</sup> and 10<sup>80</sup> 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<sup>−40</sup>.)

=== Modern studies ===
Several authors have recently identified and pondered the significance of yet another large number, approximately [[Orders of magnitude (numbers)#10100 (one googol) to 101000|120 orders of magnitude]]. This is for example the ratio of the theoretical and observational estimates of the [[Vacuum energy|energy density of the vacuum]], which Nottale (1993)<ref>
{{cite web
 |author=L. Nottale
 |title=Mach's Principle, Dirac's Large Numbers and the Cosmological Constant Problem
 |url=http://luth2.obspm.fr/~luthier/nottale/arlambda.pdf
}}</ref> and Matthews (1997)<ref>
{{cite journal
 |author=R. Matthews
 |title=Dirac's coincidences sixty years on
 |journal=[[Astronomy & Geophysics]]
 |year=1998
 |volume=39 |issue=6|pages=19–20
 |bibcode=
 |doi=10.1093/astrog/39.6.6.19
|doi-access=free
 }}</ref> associated in an LNH context with a scaling law for the [[cosmological constant]]. [[Carl Friedrich von Weizsäcker]] identified 10<sup>120</sup> 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 [[bit]]s of [[information]] in the universe.<ref>
{{cite arXiv
 |author=H. Lyre
 |year=2003
 |title=C. F. Weizsäcker's Reconstruction of Physics: Yesterday, Today and Tomorrow
 |eprint=quant-ph/0309183
}}</ref>
Valev (2019)<ref name=":0">{{Cite journal |author=D. Valev |year=2019 |title=Evidence of Dirac large numbers hypothesis |journal=Proceedings of the Romanian Academy |volume=20 |issue=+4|pages=361–368|url=https://acad.ro/sectii2002/proceedings/doc2019-4/06-Valev.pdf}}</ref> 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, {{nowrap|32.8·10<sup>120</sup>}}. This provides evidence of the Dirac large numbers hypothesis by connecting the macro-world and the micro-world.

== See also ==
{{Portal|Physics}}
* {{annotated link|Dimensionless physical constant}}
* {{annotated link|Hierarchy problem}}
* {{annotated link|Time-variation of fundamental constants}}

== References ==
{{reflist|2}}

== Further reading ==
*{{cite journal
 |author=P. A. M. Dirac
 |year=1938
 |title=A New Basis for Cosmology
 |journal=[[Proceedings of the Royal Society of London A]]
 |volume=165 |issue=921 |pages=199–208
 |bibcode=1938RSPSA.165..199D
 |doi=10.1098/rspa.1938.0053
|doi-access=
 }}
*{{cite journal
 |author=P. A. M. Dirac
 |year=1937
 |title=The Cosmological Constants
 |journal=[[Nature (journal)|Nature]]
 |volume=139 |issue=3512 |pages=323
 |bibcode=1937Natur.139..323D
 |doi=10.1038/139323a0
|s2cid=4106534
 }}
*{{cite journal
 |author=P. A. M. Dirac
 |year=1974
 |title=Cosmological Models and the Large Numbers Hypothesis
 |journal=[[Proceedings of the Royal Society of London A]]
 |volume=338 |issue=1615 |pages=439–446
 |bibcode=1974RSPSA.338..439D
 |doi=10.1098/rspa.1974.0095
|s2cid=122802355
 }}
*{{cite journal
 |author1=G. A. Mena Marugan
 |author2=S. Carneiro
 |year=2002
 |title=Holography and the large number hypothesis
 |journal=[[Physical Review D]]
 |volume=65 |issue=8 |page=087303
 |arxiv=gr-qc/0111034
 |bibcode=2002PhRvD..65h7303M
 |doi=10.1103/PhysRevD.65.087303
|s2cid=119452710
 }}
*{{cite journal
 |author1=C.-G. Shao
 |author2=J. Shen
 |author3=B. Wang
 |author4=R.-K. Su
 |title=Dirac Cosmology and the Acceleration of the Contemporary Universe
 |year=2006
 |journal=[[Classical and Quantum Gravity]]
 |volume=23 |issue=11 |pages=3707–3720
 |arxiv=gr-qc/0508030
 |bibcode=2006CQGra..23.3707S
 |doi=10.1088/0264-9381/23/11/003
|s2cid=119339090
 }}
*{{cite arXiv
 |author1=S. Ray
 |author2=U. Mukhopadhyay
 |author3=P. P. Ghosh
 |year=2007
 |title=Large Number Hypothesis: A Review
 |class=gr-qc
 |eprint=0705.1836
}}
*{{cite journal
 |author1=A. Unzicker
 |year=2009
 |title=A Look at the Abandoned Contributions to Cosmology of Dirac, Sciama and Dicke
 |journal=[[Annalen der Physik]]
 |volume=18 |issue=1 |pages=57–70
 |arxiv=0708.3518
 |bibcode=2009AnP...521...57U
 |doi=10.1002/andp.20095210108
|s2cid=11248780
 }}

== External links ==
*[https://web.archive.org/web/20101226111524/http://www.paricenter.com/library/download/dirac01.mp3 Audio of Dirac talking about the large numbers hypothesis]
*[https://web.archive.org/web/20080509071941/http://www.fdavidpeat.com/interviews/dirac.htm Full transcript of Dirac's speech.]
*[https://web.archive.org/web/20080203133606/http://ourworld.compuserve.com/homepages/rajm/agdirac.htm Robert Matthews: Dirac's coincidences sixty years on]
*[http://www.jgiesen.de/astro/stars/diracnumber.htm The Mysterious Eddington–Dirac Number]

{{DEFAULTSORT:Dirac Large Numbers Hypothesis}}
[[Category:Physical cosmology]]
[[Category:Obsolete scientific theories]]
[[Category:Paul Dirac|Large Numbers Hypothesis]]
[[Category:Astronomical hypotheses]]
[[Category:1937 introductions]]
[[Category:Coincidence]]