The Dirac large numbers hypothesis (LNH) is an observation made by
Paul Dirac
Paul Adrien Maurice Dirac ( ; 8 August 1902 – 20 October 1984) was an English mathematician and Theoretical physics, theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for bot ...
in 1937 relating ratios of size scales in the
Universe
The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
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
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...
with these unusual features:
*The strength of gravity, as represented by the
gravitational constant
The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
, is inversely proportional to the
age of the universe
In physical cosmology, the age of the universe is the cosmological time, time elapsed since the Big Bang: 13.79 billion years.
Astronomers have two different approaches to determine the age of the universe. One is based on a particle physics ...
:
*The mass of the universe is proportional to the square of the universe's age:
.
*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.
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
Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
(1919), 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:
:
where,
:
:
with
and ''r''
e is the
classical electron radius
The classical electron radius is a combination of fundamental Physical quantity, physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic sel ...
, ''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
Sir Arthur Stanley Eddington, (28 December 1882 – 22 November 1944) was an English astronomer, physicist, and mathematician. He was also a philosopher of science and a populariser of science. The Eddington limit, the natural limit to the lu ...
(1931) who related the above ratios to N, the estimated number of charged particles in the universe, with the following ratio:
:
.
In addition to the examples of Weyl and Eddington, Dirac was also influenced by the
primeval-atom hypothesis of
Georges Lemaître
Georges Henri Joseph Édouard Lemaître ( ; ; 17 July 1894 – 20 June 1966) was a Belgian Catholic priest, theoretical physicist, and mathematician who made major contributions to cosmology and astrophysics. He was the first to argue that the ...
, 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
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physi ...
. 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:
:
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:
:
where ''t'' is the age of the universe,
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
and ''r''
e is the classical electron radius. Hence, in units where and , the age of the universe is about 10
40 units of time. This is the same
order of magnitude
In a ratio scale based on powers of ten, the order of magnitude is a measure of the nearness of two figures. Two numbers are "within an order of magnitude" of each other if their ratio is between 1/10 and 10. In other words, the two numbers are ...
as the ratio of the
electrical
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
to the
gravitational
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force be ...
force
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
s between a
proton
A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
and an
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
:
:
Hence, interpreting the
charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* '' Charge!!'', an album by The Aqu ...
of the
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
, the
mass
Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
es
and
of the proton and electron, and the permittivity factor
in atomic units (equal to 1), the value of the
gravitational constant
The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
is approximately 10
−40. Dirac interpreted this to mean that
varies with time as
. Although
George Gamow
George Gamow (sometimes Gammoff; born Georgiy Antonovich Gamov; ; 4 March 1904 – 19 August 1968) was a Soviet and American polymath, theoretical physicist and cosmologist. He was an early advocate and developer of Georges Lemaître's Big Ba ...
noted that such a temporal variation does not necessarily follow from Dirac's assumptions, a corresponding change of ''G'' has not been found.
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
In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
a gauge function 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
Geophysics () is a subject of natural science concerned with the physical processes and Physical property, properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct i ...
, for instance,
Edward Teller
Edward Teller (; January 15, 1908 – September 9, 2003) was a Hungarian and American Theoretical physics, theoretical physicist and chemical engineer who is known colloquially as "the father of the hydrogen bomb" and one of the creators of ...
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
Paleontology, also spelled as palaeontology or palæontology, is the scientific study of the life of the past, mainly but not exclusively through the study of fossils. Paleontologists use fossils as a means to classify organisms, measure geolo ...
data. However,
George Gamow
George Gamow (sometimes Gammoff; born Georgiy Antonovich Gamov; ; 4 March 1904 – 19 August 1968) was a Soviet and American polymath, theoretical physicist and cosmologist. He was an early advocate and developer of Georges Lemaître's Big Ba ...
demonstrated in 1962 how a simple revision of the parameters (in this case, the age of the
Solar System
The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...
) 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
Robert Henry Dicke (; May 6, 1916 – March 4, 1997) was an American astronomer and physicist who made important contributions to the fields of astrophysics, atomic physics, cosmology and gravity. He was the Albert Einstein Professor in Scien ...
in 1961. Known as the
anthropic coincidence or
fine-tuned universe
The fine-tuned universe is the hypothesis that, because "life as we know it" could not exist if the fundamental physical constants, constants of nature – such as the electron charge, the gravitational constant and others – had been even ...
, it simply states that the large numbers in LNH are a necessary coincidence for intelligent beings since they parametrize
fusion of
hydrogen
Hydrogen is a chemical element; it has chemical symbol, symbol H and atomic number 1. It is the lightest and abundance of the chemical elements, most abundant chemical element in the universe, constituting about 75% of all baryon, normal matter ...
in
star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s and hence carbon-based
life
Life, also known as biota, refers to matter that has biological processes, such as Cell signaling, signaling and self-sustaining processes. It is defined descriptively by the capacity for homeostasis, Structure#Biological, organisation, met ...
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 10
60, an interesting variation on the 10
40 and 10
80 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
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the el ...
, 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
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is a coefficient that Albert Einstein initially added to his field equations of general rel ...
.
Carl Friedrich von Weizsäcker
Carl Friedrich Freiherr von Weizsäcker (; 28 June 1912 – 28 April 2007) was a German physicist and philosopher. He was the longest-living member of the team which performed nuclear research in Nazi Germany during the Second World War, un ...
identified 10
120 with the ratio of the universe's volume to the volume of a typical
nucleon
In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number.
Until the 1960s, nucleons were thought to be ele ...
bounded by its
Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle, defined as the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It was introduced by Arthur Compton in 1 ...
, and he identified this ratio with the sum of elementary events or
bit
The bit is the most basic unit of information in computing and digital communication. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented as ...
s of
information
Information is an Abstraction, abstract concept that refers to something which has the power Communication, to inform. At the most fundamental level, it pertains to the Interpretation (philosophy), interpretation (perhaps Interpretation (log ...
in the universe.
[
]
Valev (2019)
found an equation connecting cosmological parameters (for example density of the universe) and
Planck units
In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: ''Speed of light, c'', ''Gravitational constant, G'', ''Reduced Planck constant, ħ ...
(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, . This provides evidence of the Dirac large numbers hypothesis by connecting the macro-world and the micro-world.
See also
*
*
*
References
Further reading
*
*
*
*
*
*
*
External links
Audio of Dirac talking about the large numbers hypothesis
{{DEFAULTSORT:Dirac Large Numbers Hypothesis
Physical cosmology
Obsolete scientific theories
Large Numbers Hypothesis
Astronomical hypotheses
1937 introductions
Coincidence