11 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Dirac large numbers hypothesis | 1/2 | https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis | reference | science, encyclopedia | 2026-05-05T09:33:55.823362+00:00 | kb-cron |
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
∝
1
/
t
{\displaystyle G\propto 1/t\,}
The mass of the universe is proportional to the square of the universe's age:
M
∝
t
2
{\displaystyle M\propto t^{2}}
. 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 (1919), 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:
R
U
r
e
≈
r
H
r
e
≈
4.1666763
⋅
10
42
≈
10
42.62
…
,
{\displaystyle {\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
e
=
e
2
4
π
ϵ
0
m
e
c
2
≈
2.81794032
⋅
10
−
15
m
{\displaystyle r_{\text{e}}={\frac {e^{2}}{4\pi \epsilon _{0}\ m_{\text{e}}c^{2}}}\approx 2.81794032\cdot 10^{-15}\mathrm {m} }
r
H
=
e
2
4
π
ϵ
0
m
H
c
2
≈
1.1741445
⋅
10
28
m
{\displaystyle r_{\text{H}}={\frac {e^{2}}{4\pi \epsilon _{0}\ m_{\text{H}}c^{2}}}\approx 1.1741445\cdot 10^{28}\,\mathrm {m} }
with
m
H
c
2
=
G
m
e
2
r
e
{\displaystyle m_{\text{H}}c^{2}={\frac {Gm_{\text{e}}^{2}}{r_{\text{e}}}}}
and 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) who related the above ratios to N, the estimated number of charged particles in the universe, with the following ratio:
e
2
4
π
ϵ
0
G
m
e
2
≈
4.1666763
⋅
10
42
≈
N
{\displaystyle {\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. 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
=
(
c
3
M
U
)
t
,
{\displaystyle G=\left(\!{\frac {c^{3}}{M_{\text{U}}}}\!\right)t,}
where MU 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:
c
t
r
e
≈
3.47
⋅
10
41
≈
10
42
,
{\displaystyle {\frac {c\,t}{r_{\text{e}}}}\approx 3.47\cdot 10^{41}\approx 10^{42},}
where t is the age of the universe,
c
{\displaystyle 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:
e
2
4
π
ϵ
0
G
m
p
m
e
≈
10
40
.
{\displaystyle {\frac {e^{2}}{4\pi \epsilon _{0}Gm_{\text{p}}m_{\text{e}}}}\approx 10^{40}.}
Hence, interpreting the charge
e
{\displaystyle e}
of the electron, the masses
m
p
{\displaystyle m_{\text{p}}}
and
m
e
{\displaystyle m_{\text{e}}}
of the proton and electron, and the permittivity factor
4
π
ϵ
0
{\displaystyle 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
{\displaystyle G}
varies with time as
G
≈
1
/
t
{\displaystyle G\approx 1/t}
. Although George Gamow 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 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).