60
J. Maruani
Table 3.1 The lepton series
of the electron family
Name
Rank
E/MeV
log(E/E e )
Ratio
Electron
1
0.5110
0.
“
Muon
2
105.7
2.316
“
Tau
3
1 777
3.541
1.529
‘Next’
4
122.5 × 10 3
5.380
1.519
Planck limit
Limit
1.221 × 10 22
22.378
Limit
Fig. 3.1 Quadratic fit of the
logarithms of reduced
energies in the electron family
In our earlier paper [22] we noted that the Compton diameter (reduced wavelength): ¯
λ C ≡ λ C /2π = 2r C ∼ 3.86159 × 10 −13 m, is a geometric average of the
classical electrostatic radius: r 0 = k e e 2 /m 0 c 2 ∼ 2.81794 × 10 −15 m, and the Bohr
hydrogen radius: a 0 = 2 /k e m e e 2 ∼ 5.29177 × 10 −11 m, k e being the Coulomb
constant (k e = 1/4πε 0 ):
2r C /a 0 = r 0 /2r C = α, α = k e e
2 /c ∼ 0.729735 × 10
−2 .
(3.22)
The Compton diameter 2r C was also shown to be a geometric average of the
space-time curvatures, defined from general relativity theory, ‘inside’ the electron: r G = (G/c 2 )m 0 , and ‘outside’ a volume of radius r Q : R G = r 2
Q /r G [22]. For
r Q = r C , one is led to define a gravitational invariant δ similar to the fine-structure
constant α:
2r C /4R G = r G /2r C = δ, δ = Gm
2
0 /c ∼ 1.751 × 10
−45 .
(3.23)
Auxiliary relations resulting from Eqs. (3.22) and (3.23) can be written:
r G /r 0 = δ/α,
r G /2r C = δ,
r G /a 0 = δ · α.
(3.24)
The gravitational invariant δ introduced here differs from that, α p , introduced by
Carr and Rees [33] while discussing cosmological issues raised by Dirac, Dicke,
Jordan, and others: α p involves the mass of the proton, m p , instead of that of the
J. Maruani
Table 3.1 The lepton series
of the electron family
Name
Rank
E/MeV
log(E/E e )
Ratio
Electron
1
0.5110
0.
“
Muon
2
105.7
2.316
“
Tau
3
1 777
3.541
1.529
‘Next’
4
122.5 × 10 3
5.380
1.519
Planck limit
Limit
1.221 × 10 22
22.378
Limit
Fig. 3.1 Quadratic fit of the
logarithms of reduced
energies in the electron family
In our earlier paper [22] we noted that the Compton diameter (reduced wavelength): ¯
λ C ≡ λ C /2π = 2r C ∼ 3.86159 × 10 −13 m, is a geometric average of the
classical electrostatic radius: r 0 = k e e 2 /m 0 c 2 ∼ 2.81794 × 10 −15 m, and the Bohr
hydrogen radius: a 0 = 2 /k e m e e 2 ∼ 5.29177 × 10 −11 m, k e being the Coulomb
constant (k e = 1/4πε 0 ):
2r C /a 0 = r 0 /2r C = α, α = k e e
2 /c ∼ 0.729735 × 10
−2 .
(3.22)
The Compton diameter 2r C was also shown to be a geometric average of the
space-time curvatures, defined from general relativity theory, ‘inside’ the electron: r G = (G/c 2 )m 0 , and ‘outside’ a volume of radius r Q : R G = r 2
Q /r G [22]. For
r Q = r C , one is led to define a gravitational invariant δ similar to the fine-structure
constant α:
2r C /4R G = r G /2r C = δ, δ = Gm
2
0 /c ∼ 1.751 × 10
−45 .
(3.23)
Auxiliary relations resulting from Eqs. (3.22) and (3.23) can be written:
r G /r 0 = δ/α,
r G /2r C = δ,
r G /a 0 = δ · α.
(3.24)
The gravitational invariant δ introduced here differs from that, α p , introduced by
Carr and Rees [33] while discussing cosmological issues raised by Dirac, Dicke,
Jordan, and others: α p involves the mass of the proton, m p , instead of that of the