When we define a precise value for the gravitational constant, we can then write a precise equation for the electron Compton wavelength. This will allow us to show how the electron mass is quantized. The G value defined is within the range of values found with laboratory (mass and force) measurements.

G= (Le/4pi)^3(1/2pi)^2(1/3m)

G= 6.67174557x10^-11

The Le value is the electron Compton wavelength. The m value is the electron mass. When the values from NIST are used, the gravitational constant (G) will have the value shown above.

Le=4pi (3pi hG/c)^1/4=2.426310217x10^-12

These equations are developed when the electron is analyzed as a gravitationally collapsed, single photon geon (as proposed by Alexander Burinskii). Some new ideas are brought to the table with this analysis. One new idea is that the maximum energy photon has energy that is less than the Planck mass energy. The maximum energy photon wavelength has the energy value (2/3)^1/2 times the Planck mass energy. This photon wavelength labeled (L1) has the energy needed to produce two mass particles, each with a photon sphere circumference (or photon orbit circumference) equal to the photon wavelength. This photon is unique because its energy is defined by either the Planck constant or the gravitational constant.

E=h(frequency)=hc/L1

E=L1(c^4)(1/3pi G)

L1=(3pi hG/c^3)^1/2

L1=2pi(Planck length)(3/2)^1/2

When the two energy values (E) are equal, the (L1) wavelength is 2pi(Planck length)(3/2)^1/2. The second energy equation (above) comes from the gravitational photon orbit radius equation.

Photon orbit radius=3Gm/c^2=L1/2pi

(L1/2pi)(c^4/3G)=mc^2=(1/2)E

E=L1(c^4)(1/3pi G)

This (L1) photon has the energy density to produce "diffractive limit" space curvature so its energy has reached a limit value. The smallest meaningful time interval that relates to this photon wavelength is (Planck time)(3/2)^1/2. The smallest length is the radius value (Planck length)(3/2)^1/2.

In this model,the electron is a single photon geon, gravitationally collapsed. The photon is confined as a ring singularity at the electron radius (3Gm/c^2). It must turn through two revolutions to complete a spin cycle so a circumference is equal to one half of a (geon) photon wavelength. The length 4pi(3Gm/c^2) will then be equal to a gravitationally collapsed photon wavelength.

The known electron angular momentum value implies a radius equal to the electron Compton wavelength divided by (4pi). With this radius, electron mass times light velociy times (Le/4pi) is equal to the angular momentum value (h/4pi). The linear momentum of an electron Compton wavelength photon confined in a closed ring will produce the correct angular momentum when the ring radius is Compton wavelength divided by (4pi).

Angular momentum=(E/c)(Le/4pi)

Angular momentum=(hc/Le)(1/c)(Le/4pi)

Angular momentum=h/4pi

The electron radius is known to be much smaller than its Compton wavelength divided by (4pi) and yet, with a smaller radius, the electron would seem to have less angular momentum than the value (h/4pi). This has led a number of theorists to analyze the electron as a gravitationally collapsed (and gravitationally confined) entity. With gravitational collapse, angular momentum is conserved. There is no other way known that will allow the required angular momentum value to exist with a particle that is far smaller than the radius (Le/4pi). The (Le/4pi) value is one half of the "reduced" electron Compton wavelength value (Le/2pi).

When the electron is gravitationally collapsed, with maximal angular momentum, its photon path bending (electromagnetic) force and its (self) gravitational force will be precisely balanced. The (required) diffractive limit space curvature is predicted at the electron mass photon orbit radius (3Gm/c^2). The gravitational time dilation factor (or photon blue-shift factor) and the equal space contraction factor that will reduce the electron Compton wavelength to the size (4pi) times (3Gm/c^2) is determined next.

We find that the ratio of the length 4pi(3Gm/c^2) to the electron Compton wavelength is equal is equal to the dimensionless ratio, (3/2)^1/2(Planck time in seconds) divided by (2pi seconds).

4pi(3Gm/c^2)/Le=(3/2)^1/2(Planck time/2pi seconds)

4pi(3Gm/c^2)/Le=1.05068319x10^-44

[4pi(3Gm/c^2)/Le]^1/2=[(3/2)^1/2(Planck time)/(2pi seconds)]^1/2

[4pi(3Gm/c^2)/Le]^1/2=1.02502838x10^-22

The ratio 1.02502838x10^-22 to one is the gravitational time dilation factor found to be applicable at the electron photon orbit radius (3Gm/c^2). When the (equal) gravitational space contraction factor is included, the size ratio is 1.05068319x10^-44 to one. This is the ratio 4pi(3Gm/c^2) divided by the electron Compton wavelength.

When the radius value (Le/4pi) is reduced by the gravitational blue-shift factor 1.02502838x10^-22 to one, the new radius is (3/2)^1/2 (Planck length). In this model, this is the smallest meaningful length that can be defined without including the gravitational space contraction effect.

(Le/4pi)[(3/2)^1/2(Planck time)/(2pi seconds)]^1/2=(3/2)^1/2(Planck length)

(Le/4pi)[(3/2)^1/2(Planck time)/(2pi seconds)]=3Gm/c^2

The smallest meaningful length that relates to the electron collapsed mass is the photon orbit radius (3Gm/c^2). This size is the result of gravitational time dilation (blue-shift) and space contraction. The electron Compton wavelength is defined next.

Le/4pi=(3/2)^1/2(Planck length)[(2pi seconds)/(3/2)^1/2(Planck time)]^1/2

Le/4pi=(3pi hG/c)^1/4

Le=4pi(3pi hG/c)^1/4=2.426310217x10^-12

The photon wavelength that has electromagnetic energy equal to the mass energy of one electron plus one positron is (Le/2).

Le/2=2pi(3pi hG/c)^1/4

An equation using the (Le/2) photon wavelength energy is evaluated next. In the energy equation, the (E2) energy is 2(mc^2) where (m) is the electron mass. The (E1) value is (2/3)^1/2 (Planck mass energy) identified earlier as the upper photon energy limit. The value (E3) is determined next.

E2/E1=E3/E2

E3=(E2/E1)(E2)=1.678402875x10^-35 joule

E3=h(1/2pi)^2

E2=(E1)^1/2(E3)^1/2

E2=2mc^2=2(hc/Le)

The very small (E3) energy is found to have the value (Planck constant) divided by (2pi)^2. This is the tiny amount of energy that the (L1) wavelength photon would have if its energy is degraded (reduced) by the dimensionless time dilation factor [(3/2)^1/2(Planck time in seconds) divided by (2pi seconds)]. We will label this factor (Tf).

E3=E1(Tf)=h(1/2pi)^2

(Tf)^1/2=E2/E1=E3/E2=(E3/E1)^1/2

(Tf)^1/2=[4pi(3Gm/c^2)/Le]^1/2

(Tf)^1/2=[h/(2pi)^2]/(2mc^2)

(Tf)^1/2=1.025028384x10^-22

(Tf)=E3/E1=4pi(3Gm/c^2)/Le

(Tf)=1.050683188x10^-44

A length ratio equation clearly links the electron radius (3Gm/c^2) and the electron Compton wavelength to the radius (3/2)^1/2(Planck length). We will define (R2) along with (R4) and then solve for (R1) below.

R2=Le/4pi=(h/mc)/(4pi)=h/4pi mc

R4=3Gm/c^2

R1/R2=R4/R1

(R1)^2=(R2)(R4)=(h/4pi mc)(3Gm/c^2)

(R1)^2=(3hG/4pi c^3)

(R1)=(3hG/4pi c^3)^1/2=(3/2)^1/2(Planck length)

This last (R1) equation is clear and precise. The equation that defines the electron Compton wavelength is related to (R4) and (R1) by the dimensionless time dilation factor (Tf).(with units second per second.)

4pi(R4)(1/Tf)=(Le)=Compton wavelength

4pi(R1)(1/Tf)^1/2=(Le)=Compton wavelength

4pi(3/2)^1/2(Planck length)(1/Tf)^1/2=4pi(3pi hG/c)^1/4

4pi(3pi hG/c)^1/4=(Le)=electron Compton wavelength

The electron mass is a quantized value when its Compton wavelength is quantized.

We have defined equations that work but we don't have established theory that is compatible. Some theorists, including Leonard Susskind, have anticipated that the electron radius is not much bigger, or not much smaller than the Planck length. Other theorists who are convinced that the electron is a fundamental particle with no internal structure, will view the ratio relationships described above as nothing more than numerology. Properties implied from equations developed include a fixed time dilation factor applicable at the photon sphere radius that is independent of a specific mass value. This may be helpful to explain the muon, with its larger mass photon sphere and yet, smaller magnetic moment.

Those who expect internal structure, will find there can be little doubt remaining that the electron is a gravitationally confined entity. I advised Alexander Burinskii March 30, 2009 that the electron mass code has been broken.

Electron mass=(h/4pi c)(c/3pi hG)^1/4

Much of the work needed to show that a self-confined, single wavelength photon, has the fundamental properties of an electron was done by J.G.Williamson and M.B.van der Mark. Their paper "Is the electron a photon with toroidal topology?" (1997) shows that the electron charge can arise from the toroidal topology of the photon path, in combination with the photon electric field. Their paper describes the electron as a toroidal ring with a ring radius equal to a single photon wavelength divided by (4pi). A left-handed flow produces an electron while a right-handed flow produces a positron. When the (John Wheeler) "gravitational collapse" suggestion is included in this concept, the new model defines the quantized electron mass value with a specific relationship to the Planck mass.

Electron mass=(2/3)^1/2(Planck mass)(1/2)(Tf)^1/2

In this equation, the time dilation factor (Tf) is dimensionless so when the Planck mass is specified in kilograms the electron mass is specified in kilograms. These relationships imply that the Planck mass value is misleading because we will not expect to find any single mass particle (or photon) that has energy (mc^2) equal to the Planck energy.

In this model, the new idea of a maximum energy (minimum size) photon wavelength is significant. The applicable gravitational time dilation factor is then easily determined.