The Angular Momenta Dipole Moments and Gyromagnetic Ratios of the Electron and the Proton ()
1. Introduction
The purpose of this work is to obtain analytical formulae for the dipole moments and angular momenta of the electron and the proton. For this purpose, exact solutions of the Einstein-Maxwell field equations for volume distributions of rotating charged matter are required. There are difficulties in obtaining such solutions, but these were successfully treated in detail in [1] . In addition, there are uncertainties regarding the radii of these particles, resulting in some differences between the calculated numerical values and the accepted values of the dipole moments.
2. The Electron
The mass
, classical radius
, the electric charge 
of the electron and the vacuum speed of light, are:
(1)
The following ratios are required:
(2)
If the mass is
, then there is an electric charge 
associated with 
[1] . Thus, if there is an additional electric charge
, the total charge is
. The value of 
is only 
and so it is negligible compared to the value of 
in (1). If
, we may therefore write for the charge 
and its square
,
(3)
In the Reissner-Nordstrom solution [2] , the coefficient of 
in the metric, has the form given by
(4)
with 
given by the second of (3).
The coefficient of 
in the Kerr-Newman metric, is
(5)
with
. (6)
Equation (5) then, becomes
(7)
where 
is the angular momentum per unit mass; see Equations (18)-(20) of [2] . On
, 
and so on 
Equation (7) becomes
(8)
which is the same as Equation (2) of [2] (Note that in reference [2] 
is used where we use
). It follows that instead of Equations (7)-(8) of [1] , we have for 
(9)
In accordance with the results of [1] , the dipole moment
, total angular momentum 
and gyromagnetic ratio are:
(10)
where
(11)
The values in (2) and (3) give for 
in (9) and 
(12)
Equations (10) then give
, or
(13)
The accepted value 
is
. (14)
From the second of (10), we obtain for 
. (15)
The gyromagnetic ratio is therefore,
(16)
and this is precisely the value of
.
It must be noted that there are various different values for the radius of the classical electron. The value of 
found is smaller than the accepted value and it depends on which radius we choose. Thus, if we use a radius with numerical value
(17)
which is 7 times larger than the classical radius of the electron and repeat the calculations, we find
(18)
This is nearer the accepted value in (14).
It was established by Dehmelt [3] that the upper limit for the electron radius is
.
This simply implies that if 
is the electronic radius, we must have
. The values of the radii we used, including the “classical radius of the electron”, are much larger than this.
3. The Proton
The mass, radius, electric charge and mass to radius ratio of the proton, are respectively:
(19)
If there is an additional electric charge
, the total charge will be
. The value of 
is only 
and so it is negligible compared to the value of 
in (1). If
, we may therefore write for the total charge 
and its square
,
(20)
The same Formulae (12) for 
and (10) for
, 
and 
are valid, but with the proton parameters in Equations (19) instead of the electron ones. We find
(21)
We note that the gyromagnetic ratio found in the 
of Equations (21), is precisely the value of
.
In the case of the proton the value of 
found is larger than the accepted value, which is
.
4. Conclusion
We have obtained the dipole moments angular momenta and gyromagnetic ratios of the electron and the proton using the analytical formulae developed in [1] . In the case of the electron, the value of 
found is smaller than the accepted value, but in the proton case it is larger. If Dehmelt’s [3] deductions are valid, a complete reevaluation is necessary using our analytical Formulae (10) to find
, 
and
. It is not possible to do this, because Dehmelt, does not give any definite values for the radii; he only states that the electron radius 
should satisfy the inequality
. But in any case, the purpose of the calculations here, is to see if the values of the known classical electron and proton radii, give the expected values of
, 
and
.