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 and gravitationally confined entity (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 radius equal to (L1) divided by 2pi. 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 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) of photon energy
E = L1 (c^4) (1/ 3pi G) = photon energy
This (L1) photon has the energy density to produce "diffractive limit" space curvature so its energy has reached a critical, limit value. The electromagnetic and gravitational forces become equal when a photon has the L1 wavelength 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 charge is confined within a ring at the electron radius (3Gm/ c^2). It must turn through two revolutions to complete a spin cycle so a circumference is equivalent to one half of an electron Compton wavelength. The length 4pi (3Gm/ c^2) will then be equivalent to an electron matter wavelength, or de Broglie 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 velocity 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 elecron 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 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 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)]. This factor, labeled (Tf) is defined as the gravitational time dilation limit.
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). (Units are 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 predictable gravitational time dilation factor or energy degradation factor that applies at the photon orbit radius for each of the electron, muon and tau particles. The three specific energy degradation factors explain the three different mass values that we observe for the electron muon and tau particles. This is helpful to explain how the muon, and tau particles each have a larger photon sphere radius and yet, have smaller magnetic moment values than electrons have.
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
When the (John Wheeler) "gravitational collapse" suggestion is incorporated, the electron model defines a 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
Electron mass = (hc/12 pi G)^1/2 (3hG/4pi c^5)^1/4 (1/2pi)^1/2
Electron mass = 9.109382147x10^-31 kg
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 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. Materializing an electron, positron pair requires that a photon with energy equal to or greater than 2mc^2 must be blue shifted to the wavelength labeled (L1) in the gravitational field of an existing particle. The blue shifted photon then has the energy density required for self confinement in a one wavelength loop. The photon closed loop will consist of two energy packets, one with positive electric field energy and one with negative electric field energy. Each energy packet will become a mass particle when its electromagnetic energy is converted to gravitational field energy (when each packet has a time dilation factor of zero seconds per second as compared to a distant observer). The following conditions will then apply:
The electron ring will be just within the static limit of the spinning electron gravitational field. This ring is in the ergosphere region, where the inertial frame is expected to spin at light velocity and the time rate (at radius, 3Gm/c^2) is zero seconds per second. The electron charge field can then appear to be stationary to an outside observer and yet have the spin acceleration needed to have the charge voltage equal to 0.511 MeV, due to Unruh effect along with inertial frame spin.
Some summary equations are useful to show units that are used. Length values are meters, and electron mass (m) is kg. One second is the time required for light to travel 299792458 meters while the Planck time is the time required for light to travel one Planck length.
Re = 2Gm/ c^2 = electron Schwarzschild radius
Re = 2/3 (Le/ 4pi) (Le/2L3)^2
Le = 2.4263102175x10^-12 meters
L3 = (2pi seconds) (c) (2pi) = 1.183533185x10^10 meters
Re = 2/3 (Le/ 4pi) (Le/ 2L3)^2 = 1.352436826x10^-57 meters
G = (c^2/ 2m) (1.352436826x10^-57)
G = 6.67174557x10^-11
This G value is consistant with the 1986 CODATA value, 6.67259x10^-11, with standard uncertainty 0.00085x10^-11. When the standard uncertainty is subtracted, the G value is 6.67174x10^-11. The larger 2006 CODATA value for G is 6.67428x10^-11. See:
The well known theorist P. A. M. Dirac has stated, the aim is "not so much to get a model of the electron as to get a simple scheme of equations which can be used to calculate all the results that can be obtained from experiment." The simple equations presented here provide a step toward achieving this.
This model will be reviewed with information added when appropriate. Review cycle is expected to be two times per year. Information was added Aug. 28, 2010.
Information was added June 17, 2011, relating to muon and tau particles. Information was added June 26, 2011 to clarify time dilation.
On this day Aug. 8, 2011, information is added to incorporate the new CODATA value for the electron Compton wavelength shown as Le.
Le = 4pi (3pi hG/c)^1/4 = 2.4263102389x10^-12 meter
(Le/4pi)^4 (c/3pi) = (h) (G) = 4.420744783x10^-44
The latest CODATA value for the Planck constant is 6.62606957x10^-34. Five separate methods have been used to determine the Planck constant. This is a weighted mean of values calculated by CODATA. A value for the gravitational constant is determined as shown.
(Le/4pi)^4 (c/3pi) (1/h) = G
G = (4.420744783x10^-44) / (6.62606957x10^-34)
G = 6.671745197x10^-11
This G value is now used in the equation previously developed to determine electron mass.
m = (1/3G) (Le/4pi )^3 (1/2pi)^2
m = 9.109382904x10^-31 kg
The latest CODATA value for electron mass is 9.10938291x10^-31 kg. The electron mass will also be determined from the electron Compton wavelength as shown.
m = (h/Le) (1/c)
m = [(6.62606957x10^-34)/(2.4263102389x10^-12)] (1/c)
m = 9.109382903x10^-31 kg
The new CODATA values for electron Compton wavelength, Planck constant and electron mass are consistant with the new derived G value.
G = 6.671745197x10^-11
The maximum mass for a single fundamental charge particle is found to be (hc/12pi G)^1/2 kg. A charge with this mass will have temperature as shown. The K value is the Boltzmann constant, 1.3806504x10^-23 joule per degree.
T = mc^2/K
T = (hc/12pi G)^1/2 (c^2/K)
T = 5.785098326x10^31 Kelvin
At any greater temperature, we find an incompatability between general relativity and quantum mechanics and so this is a maximum, upper limit temperature value. This is a smaller value than the Planck temperature which was expected to be an upper limit value.
Readers who are interested in this electron model will want to see