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The Mysterious Eddington-Dirac Number

Different relations between atomic and cosmic quantities and fundamental constants are leading to the same large number in the order of magnitude of 1040.

1. Forces

 The electrostatic force between an electron and a proton and the gravitational force
have a ratio independent of distance

= 2,27·1039

For two electrons the ratio becomes

Fc / Fg = 4,2·1042

2. Lengths

The "classic electron radius" r can be computed assuming that the energy W=mec2 is equal to the potential energy of the elementary charge e spread over a sphere of radius r:

r = 3·10-15 m

The ratio of this "elementary length" to the radius of the universe R = c·t = 1·1026 m

R / r = 3·1040

is a number of the same order of magnitude as in (1).

3. Times

The light takes the time t to pass the elementary length

t = r / c = 10-23 s

This "elementary time" is contained in age of the universe T = 6,2·1017 s by a number of the same order of magnitude as in (1) and (2):

T / t = 6·1040

4. Particles

The mass M of the universe 2,4·1051 kg to 2,0·1052 kg compared to the mass of a proton mp = 1,67·10-27 kg is the number of protons

M / mp = 6·1078

and the number of particles (protons and electrons) is

N = 1,2·1080

This is nearly the square of the number found in (1), (2) and (3) !

N = (Fe / Fg)2 = (R / r)2 = (T / t)2

By chance or not ?

Dirac suggested in 1937 that this coincidence could be understood if fundamental constants - in particular, G - varied as the Universe aged.

Robert Dicke pointed out in 1957 and 1961 that the age of the universe, as seen by living observers, cannot be random: The coincidence is is a consequence of the fact that 'carbon is required to make physicists' to observe the universe. The order of magnitude of the lifespan of a main sequence star (Sun: 10·109 years) agrees with the result derived by Dirac.

Another strange coincidence:

The ratio c2/G (square of the speed of light c divided by the gravitational constant G) is nearly the same as the the ratio M/R (mass M of the universe and radius R of the visible universe):

c2 / G  = M / R

c2/G = (2.998·108 m/s)2/[6.674·10-11 m3/(kg s2)]

c2/G = 1.4*1027 kg/m

Computing the radius R of the visible universe by c and the age T of the universe:

R = c·T
T = 13.75
·109 years = 4,34·1017 s
R = 1.30
·1026 m

and the mass M of the universe by the number of nucleons n = 1.2*1080 of mass m=1.67*10-27 kg

M = n·m = 2.00·1053 kg
M/R
= 2.00·1053 kg / 1.30·1026 m

M/R = 1.5·1027 kg/m

c2/G = M/R is equivalent to G·M/(R·c2) =1

6.674·10-11 m3/(kg s2) · 2.00·1053 kg / [1.30·1026 m · (2.998·108 m/s)2] = 1.1

------

The relation G·M/(R·c2) can be written as the ratio

of G·M·M/R and M·c2

where G·M·M/R is the gravitational potential energy for a partical of mass M in its own gravitational field, and M·c2 is the rest enrgy of mass m.

Gravitational energy and the rest energy of the universe are nearly the same !

------

Writing
R = G
·M / c2

the radius R of the universe is half of the Schwarzschild radius RS of a mass M:

RS = 2·G·M / c2

Is the universe a black hole, or a white hole ?

The expression

c2·R /M = 5,8·10-11 m3/(kg s2)

is nearly the constant of gravitation:

G = 6.67·10-11 m3/(kg s2)

 Fundamental Constants: electron charge e = 1.602·10-19 C electron mass me = 9.109·10-31 kg proton mass mp = 1.673·10-27 kg electric constant 8.854·10-12 As/(Vm) constant of gravitation G = 6.674·10-11 m3/(kg s2) speed of light (vakuum) c = 2.998·108 m/s

 Web Links Helge Kragh (2010): The Road to the Anthropic Principle Anthropic principle (Wikipedia) Dirac, P. A. M.: A New Basis for Cosmology, Proc. R. Soc. Lond. A 1938 165, 199-208, PDF Books Barrow, John D.: The Constants of Nature: The Numbers That Encode the Deepest Secrets of the Universe; Vintage Books, 2004. ISBN 978-1400032259 Rees, Martin: Just Six Numbers: The Deep Forces That Shape the Universe, Weidenfeld & Nicolson, 1999. ISBN 978-0297842972