The theorist, Alexander Burinskii has modeled the electron as a naked ring singularity. In this model, the ring singularity radius is larger than the electron Schwarzschild radius so the singularity is naked. We can add to this model and test a hypothesis that the electron is a gravitationally collapsed entity with its collapse halted at the electron photon orbit radius, 3Gm divided by c squared. The electron mass energy is equal to 1/2 of the electromagnetic energy of a photon with the wavelength 1.213155109x10^-12 meters. This photon energy is labeled (E2) in the hypothesized electron mass energy equation below.

E1/E2 = E2/E3

The E1 is defined as (2/3)^1/2 times Planck mass energy while the E3 value is (1/2pi)^2 times the energy of a photon with frequency one cycle per second.

E1 = (hc^5/3pi G)^1/2 joule
E2 = 2mc^2 joule
E3 = h/(2pi)^2 joule
E2 = (E1)^1/2 (E3)^1/2 = (hc^5/3piG)^1/4 [h/(2pi)^2]^1/2

The electron mass is E2/(2c^2). The electron mass will then be (h/4pi c)(c/3pi hG)^1/4 kilogram. Further evaluations provide evidence that this value is precisely correct.

The electron Compton wavelength, labeled (Le) is determined next.

Le = h/mc = (h/c)(1/m)
Le = (h/c)(4pi c/h)(3pi hG/c)^1/4
Le = 4pi(3pi hG/c)^1/4
(1/2)(Le) = 1.2131551x10^-12 meters = 2pi(3pi hG/c)^1/4
h/2mc = 1.213155119x10^-12 meter
(h/4pi mc)(c/3pi h)^1/4 = G^1/4
G = 6.671745178x10^-11 N m^2/kg^2

This value for G is probably the best that can presently be determined. The larger mass muon is not stable because it can not collapse to a balanced condition without radiating excess mass energy away. A balanced condition requires that time dilation outside of the particle light orbit radius will be equal to time dilation within the particle light orbit radius. The electron mass value, E2/(2c^2) implies, equal time dilation factors are required to produce a stable state. With this concept, the electron is a gravitational wave geon rather than an electromagnetic wave geon.

In October, 1954, John Wheeler and Einstein agreed that GR allowed for electromagnetic wave geon solutions. However, electromagnetic geons were found to be unstable. Wheeler proposed, if a geon is small "-- a geon might radiate away some of its energy in electron-positron pairs". See John Wheeler book,"Geons, Black Holes, And Quantum Foam", page 237-238. Wheeler writes, "Such a (gravitational wave) geon, it seemed to me, might offer a transitional state between gravity waves and a black hole". A single photon (unstable) electromagnetic geon may change into a pair of gravitational wave geons that are stable.

When the electron is described as a gravitational wave geon, it has some, but not all of the predicted properties of a black hole. The electron lives in a space where its reference frame spins at light velocity. This space is very different from the one we live in.

Don Stevens

The equaton defining gravitational attraction force between two electron mass particles separated by one electron Compton wavelength (h/mc) can be used to determine a gravitational constant value.

F = G (m)^2 /(distance)^2
F = G (m)^2 (mc/h)^2
F = G (m)^4 (c/h)^2

When electron mass is (h/4pi c)(c/3pi hG)^1/4 as found earlier, then (m)^4 has the value (h/4pi c)^4 (c/3pi hG).

F = G (h/4pi c)^4 (c/3pi hG)(c/h)^2
F = (h/3pi c)(1/4pi)^4 = G(m)^2/(distance)^2

G = (h/3pi c)(1/4pi)^4 (2.4263102175x10^-12 meter /9.10938215x10^-31 kg)^2

G = 6.67174557x10^-11

When this G value is used, the electron Compton wavelength equation value is 2 times (1.213155109x10^-12 meters). This force evaluation and derived G value strongly supports the Burinskii concept, that the electron is a gravitationally confined, naked, ring singularity, consisting of a single accelerated charge, trapped by its own spinnng inertial frame and gravitational field. This allows the charge to appear stationary to an outside observer and yet have the effective accelerated charge voltage of 0.511 MeV due to Unruh effect.

The diffraction limit space curvature, required in this concept, is found at the electron radius, 3Gm/c^2. Forces are balanced at this radius so that gravitational collapse is halted. The electron does not collapse to its Schwarzschild radius and so its ring singularity is naked. The electron is too small to measure but it does have a non-zero size so it does not have the problem of infinite, or unknown density. Malcolm Mac Gregor has said, stability considerations alone will require a non-electromagnetic force that holds the electron together. He writes,"If we are to consider an extremely small size --- then gravitational forces could be invoked to solve the (electron) stability problem." See his book, The Enigmatic Electron, 1992 (page 72). Mac Gregor provides additional reasons to show that the electron cannot be a point particle. The electron must form a quantum current loop (with spin) in order have the known magnetic moment.

The following equation may be used as an alternate method to solve for the value G. The (m) is electron mass. The L2 value is 1/2 (electron Compton wavelength).

3Gm/c^2 = (L2/2pi)^3 [1/(2pi seconds)c]^2
G = 6.67174557x10^-11

Don Stevens

I will respond to a request to show a comparison of the electron predicted mass and the NIST (experimental) electron mass value. The predicted mass will be determined, first by using the current NIST value for the gravitational constant. This G value is 6.67428x10^-11. The Planck constant (h) value used is 6.62606896x10^-34.

NIST mass = 9.10938215x10^-31 kg
Predicted mass = (h/4pi c)(c/3pi hG)^1/4 = 9.108517246x10^-31 kg

Predicted mass / NIST mass = 9.108517246x10^-31 / 9.10938215x10^-31 = 0.999905053

Further evaluations imply that the G value recommended by CODATA in 1986 is more nearly correct than the current NIST value. The 1986 value for G was 6.67259x10^-11 with standard uncertainty plus or minus 0.00085x10^-11. When this value is used for the gravitational constant, the predicted mass is shown below.

Predicted mass = (h/4pi c)(c/3pi hG)^1/4 = 9.109093931x10^-31 kg

Predicted mass / NIST mass = 9.109093931x10^-31 / 9.10938215x10^-31 = 0.99996836

This correlation is better than the first correlation. The G value required to obtain full one to one correlation is just within the standard uncertainty range specified for the
1986 CODATA value.

A recent determination of the G value using laser interferometry, dated Sept. 9, 2010 found G to be 6.67234x10^-11. See "A Simple Pendulum Determination of the Gravitational Constant", arXiv:1008.3203v3 [physics.class-ph] 7 Sep 2010 .
With this G value, the predicted electron mass is 9.109179255x10^-31 kg. This is 0.999977727 times the present CODATA electron mass value.

Don Stevens

I will respond to a request to provide background information and steps taken to define the electron mass equation. A pattern of equal length ratios was first found (without speculative explanation) that allowed the electron radius to be related to the Planck length. This pattern follows.

L1/L2=L2/L3=L4/L1=(L4/L2)^1/2=(L4/L3)^1/3=2L1/Le=Le/2L3

L1=2pi(Planck length)(3/2)^1/2=(3pi hG/c^3)^1/2
L2=1/2(electron Compton wavelength)= 1/2(Le)= h/2mc
L2= 1.213155119x10^-12
L3=(2pi)^2 (c)(one second)
L4=2pi(3Gm/c^2), where the (m) value is the electron mass

The L3 value is found from the equation:

(L2)^2=(L3)(L1)
(L2)^2=(L3)(2pi)(Planck length)(3/2)^1/2
(L2)^2 (1/L1) = L3 = 1.183533185x10^10 meter
L3 = (2pi)^2 (3pi hG/c)^1/2 (c^3/3pi hG)^1/2
L3 = (2pi)^2 (c)(one second)

The length ratios shown are all equal only when the gravitational constant has the value 6.671745178x10^-11. The L3 length will then be 1.183533185x10^10 meters. A photon with this wavelength will have the tiny energy value, hc/wavelength or hc/1.183533185x10^10 or 1.678402875x10^-35 joule. This is the energy value shown as E3 in the energy equation.

E3=h/(2pi)^2 = 1.678402875x10^-35 joule

Any reader can verify that these ratios are equal and the electron mass is (h/4pi c)(c/3pi hG)^1/4 kg when G has the value 6.671745178x10^-11.

Three factors applied to the Planck mass will provide the electron mass value. The first is (1/2), the second is (2/3)exponent 1/2, while the third is the time dilation factor applicable at the electron mass radius (3Gm/c squared). The applicable time factor is equal to the ratio (L4/L2) exponent 1/2. This is the ratio 1.025028393x10 exponent -22 to one. The E1 energy value is equal to the E3 energy, h/(2pi)^2, when it is degraded by the proposed limit time dilation factor.

Limit time factor = (3/2)^1/2 (Planck time)(1/2pi sec)

Limit time factor = (1.025028393x10^-22)^2 sec/sec

Limit time factor = 1.050683206x10^-44 sec/sec

The Planck time is the time required for light to travel one Planck length while one second, in this concept, is the time required for light to travel 299792458 meters.

(Planck length)/ c (one second) = Planck time

The Penrose Pair Production process implies that gravitational collapse, induced by a pre-existing gravitationally collapsed mass, causes captured gamma photons to annihilate and produce a plasma of electrons and positrons in the photon sphere region of the pre-existing black hole. The pre-existing black hole is required to have a large value of spin so that its photon sphere is within its ergosphere region. This is within the black hole static limit where gravitational time dilation is extreme. The ingredients of this effect are an electromagnetic field, rotating space-time and extreme (limit) photon blue-shift. When captured gamma photons exceed the rest frame energy of (approx.) one MeV, pairs of leptons are produced. Excess energy can then be radiated away as electron, or positron (mass) particles.

See "Electron and positron pair production in gravitational collapse (Oct 2011) arXiv:1110.0700v1 [astro-ph.HE]". The later version of this paper is dated March 5, 2012.

From the electron Compton wavelength and the electron mass, a gravitational constant value is derived. The Le value used is h/mc or 2.426310237x10^-12 meter.

G = (Le/4pi)^3 (1/2pi)^2 (1/3m)
G = 6.671745178x10^-11

From the electron angular momentum, electron mass and the derived gravitational constant, the correct Planck constant value is derived.

h/4pi = mc [12 (pi)^2 G m]^1/3
h = 4pi mc [12 (pi)^2 G m]^1/3
h = 6.62606957x10^-34

Since 1798 when H. Cavendish found the G value 6.754 plus or minus 0.041 (times 10 exp.-11) a range of values for G has been found. The 1986 CODATA value shown below, is most consistant with this evaluation.

G = 6.67259x10^-11 plus or minus 0.00085x10^-11

The derived G value is within the standard uncertainty range found for the 1986 G value.

The muon is found to have a relationship to the Planck mass also. Three factors applied to the Planck mass will provide the muon mass value. The first is (1/2) the second is (2/3) exponent 1/2, while the third is the square root of the ratio 4pi(3Gm/c squared) divided by (muon Compton wavelength). This factor 2.119433583x10 exponent -20 to one is the applicable time dilation factor at the muon light orbit radius.

(hc/12pi G)^1/2 (2.119433583x10^-20) = 1.88353130x10^-28 kg = muon mass

In the electron evaluation, the value, (hc/12pi G) exp 1/2 was found to be 8.886956668x10 exp-9 kg. In the muon mass equation, the G value must cancel. This requirement is satisfied only when the muon photon orbit radius, 3Gm/c squared is specified by using the consistant G value, 6.671745178x10 exp -11.

For the muon, the time dilation factor within the light orbit radius will be equal to the limit factor divided by 2.119433583x10 exponent -20 seconds per second.

(1.05068319x10^-44)/(2.119433583x10^-20) = 4.957377284x10^-25

mass energy = (1/4.957377284x10^-25)(1/2)[h/(2pi)^2] = 1.692833507^-11

muon mass = (1.692833507^-11)/c^2 = 1.883531297^-28

CODATA muon mass is 1.88353130x10^-28 kg

The muon is not stable because the time dilation factor within its light orbit radius is greater than the square root of the limit factor. The limit gravitational time dilation factor based on Planck time and the G value 6.67174557x10^-11 applies to (both) electrons and muons. This provides added confirmation that the derived G value is valid.

The tau lepton mass is related to the Planck mass by the same three factors, (1/2), (2/3)^1/2 and the square root of the ratio 4pi(3Gm/c^2) divided by the tau lepton Compton wavelength. The ratio, tau mass (3.16747x10^-28 kg) divided by (hc/12 piG)^1/2 is 3.564178781x10^-19 to one. The limit gravitational time dilation factor applies to tau leptons also. The remaining time dilation is applicable within the tau photon orbit radius, 3Gm/c^2. A minimum quantum mass value labeled (Qm) is (1/2c^2)(h/2pi)(1/2pi). This mass, 9.337375264x10^-53 kg. has the energy value 1/2[h/(2pi)^2] joule.

tau mass = [(3.564178781x10^-19)/(1.05068319x10^-44)](Qm)
tau mass = (3.392248791x10^25)(9.337375264x10^-53 kg)
tau mass = 3.16747x10^-27 kg
Limit time factor =(1/3.392248791x10^25)(3.564178781x10^-19)
Limit time factor = 1.05068319x10^-44 sec/sec

In this concept, the smallest amount of time that has meaning is (3/2)^1/2 times (Planck time). Without including gravitational length contraction, the smallest amount of length that has meaning is (3/2)^1/2 (Planck length).

The electromagnetic and gravitational forces become equal when photon frequency is (c)/ (L1). At this frequency, photon energy is equal to the energy of two fundamental charges when each charge has the spin acceleration force value (c^4/ 3G).

(radius)(force) = mc^2
(3Gm/ c^2)(c^4/ 3G) = mc^2 = (L1/2pi)(c^4/3G) = (hc/L1)(1/2)
(3Gm/ c^2)(c^2/ 3G) = m = (L1/2pi)(c^2/3G)
m = m = (hc/12pi G)^1/2 = (1/2)(2/3)^1/2 (Planck mass)

The maximum mass for a single fundamental charge particle was found to be (hc/12pi G)^1/2 or 8.886956x10^-9 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 = 5.7850979x10^31 Kelvin

At any greater temperature, we find an incompatability between general relativity and quantum mechanics and so this is found to be a maximum, upper limit temperature value. This is smaller than the Planck temperature value which is (approximately) 1.417x10^32 Kelvin.

The comparable electron temperature is:

Te = (h/4pi c)(c/3pi hG)^1/4 (c^2/K)
Te = 5.929889552x10^9 Kelvin

The ratio, electron temperature to maximum temperature is:

Te/T = 5.929889552x10^9 / 5.7850979x10^31
Te/T = 1.025028384^-22

The Te/T ratio value is the square root of the time dilation limit value found above.

Limit time factor = 1.05068319x10^-44 sec/sec
Limit time factor = (3/2)^1/2(Planck time)/2pi sec

The approximate magnetic moment values for the electron, muon and tau particles are specified by an equation developed for Kerr-Newman black holes by Brandon Carter (1968). See book, Gravitation by Misner, Thorne and Wheeler (page 891-892).

magnetic moment = (charge/mass)(angular momentum)
magnetic moment = (e/m)(h/4pi)

For the electron, this magnetic moment value is the Bohr magneton.

The electron charge acceleration is correctly determined from its angular momentum value h/4pi.

h/4pi = mcr
(h/4pi)(1/mc) = r
2pi r = Le/2
r = Le/4pi
acceleration = c^2/r = c^2 (4pi/Le)
acceleration force = mc^2/r
E = (mc^2/r)(r) = (hc)(1/Le) = mc^2
2E = (2hc)(1/Le)

As Bill Unruh (1976) has shown, charge voltage increases linearly with charge acceleration. The photon wavelength with the correct acceleration to materialize two unit charge particles, each with voltage raised to 510999 volts is (1/2) electron Compton wavelength. This is consistant with the voltage to frequency equation.

voltage/frequency = h/2e
h(frequency) = 2e (voltage)
h(frequency)(1/2e) = 510999 volts
h(c/1.21315512x10^-12)(1/2e)= 510999 volts

The voltage to frequency equation also defines an impedance value.

(1/2e)(voltage/frequency) = (1/2e)(h/2e)
(1/2e)(voltage/frequency) = h(1/2e)^2 = 6453.2 ohm

When electron capacitance is known and impedance is known then inductance is known. Capacitance is 3.1353814x10^-25 farad so that inductance must be 1.3056921x10^-17 henry.

6453.2 ohm = (inductance/capacitance)^1/2
inductance = 1.3056921x10^-17 henry
capacitance = 3.1353814x10^-25 farad
(6453.2 ohm)/(Zo) = 17.1295
Zo = 376.73 ohm
h(1/2e)^2 (8/Zo) = 1/a = 137.035999

The impedance increase from Zo to 6453.2 ohm is the result of charge acceleration.

From the book, Three Roads To Quantum Gravity by Lee Smolin, we find a discrete time and a discrete length are required in order to understand the atomic structure of geometry. See Chapter 8, from, Three Roads To Quantum Gravity. Lee Smolin provides evidence to show that space and time must be discrete. The discrete length (3/2)^1/2 (Planck length) and the discrete time (3/2)^1/2 (Planck time) are consistent with the known electron mass value.

Leonard Susskind, in his book, The Black Hole War, describes the limitation on the smallness of things as a cutoff. He writes, "Physicists have long speculated that the Planck length is the ultimate atom of space. Feynman diagrams --- make perfect sense as long as you cease adding structures smaller than the Planck length ---". (See page 335.)

The large electron angular momentum halts gravitational collapse at the radius 3Gm/c^2. With this limitation, the applicable cutoff radius dimension is found to be (3/2)^1/2 (Planck length). The cutoff photon wavelength will then be 2pi(3/2)^1/2 (Planck length). This photon wavelength is the blue shifted value that is consistent with gravitational time dilation.

Don Stevens

I will answer the question: Why is the fundamental length associated with the electron (labeled Lf) larger than the Planck length (labeled Lp)? The Planck length is the square root of the product of two radius values as shown.

Lp = [(h/2pi mc)(Gm/c^2)]exponent 1/2 = (hG/2pi c^3)^1/2

A radius value that relates to the electron is determined as shown. This is the upper limit radius. Any radius larger than this would cause angular momentum to have an unrealistic value, greater than (h/4pi).

mc(radius) = h/4pi = electron angular momentum
radius = (h/4pi mc)

The second radius value that relates to the electron is (3Gm/c^2) rather than the gravitational radius (Gm/c^2). The electron angular momentum requires a minimum radius value (3Gm/c^2). This is the photon orbit radius for the electron mass. The fundamental length associated with the electron has the value shown.

Lf = [(h/4pi mc)(3Gm/c^2)]^1/2
Lf = (3hG/4pi c^3)^1/2 = (3/2)^1/2 (hG/2pi c^3)^1/2
Lf = (3/2)^1/2 (Planck length)

The fundamental mass associated with the electron has the value shown.

Mf = (hc/12pi G)^1/2 = (1/2)(2/3)^1/2 (Planck mass)

The equation below is easily shown to be algebraically correct. This time dilation equation is numerically correct only when the derived value for G is used.

[4pi(3Gm/c^2)/(h/mc)]^1/2 = (electron mass)/(hc/12pi G)^1/2
[4pi(3Gm/c^2)/(h/mc)]^1/2 = 1.0250287x10^-22
(electron mass)/(hc/12pi G)^1/2 = 1.0250287x10^-22

I have said this time dilation (and mass ratio) equation defines the golden ratio for particles because this ratio shows up repeatedly in particle computations.

(2pi) (3/2)^1/2 (Planck length)/(h/2mc) = golden ratio
(2pi) (3hG/4pi c^3)^1/2 (1/1.213155x10^-12) = golden ratio
1.0250287x10^-22 to one = dimensionless golden ratio

[h/(2pi)^2] / (2mc^2) = 1.0250286x10^-22 = golden ratio

The last equation above defines the golden ratio very precisely because the (h) value and the (m), electron mass value, is precisely known.

(m)/(hc/12pi G)^1/2 = [h/(2pi)^2] / (2mc^2) = golden ratio
(m)(2mc^2) = [h/(2pi)^2] (hc/12pi G)^1/2
(m)^2 = (1/2c^2) [h/(2pi)^2] (hc/12pi G)^1/2
(m) = (h/4pi c) (c/3pi hG)^1/4

All electrons are identical because each electron has the mass value shown and each electron has the angular momentum value h/4pi.

In his book, "The Quantum World" (2005) author, Kenneth Ford asks, page 74, "Yet despite the incredible weakess of gravity, does it play a role at dimensions far smaller than the size of a proton? Does it become intertwined with quantum theory in ways that we can only dimly imagine? How marvelous it will be when we find out." On page 105, Ford asks, "Why is the mass of the muon more than two hundred times the mass of the electron? --- Quantized mass is just there, awaiting explanation."

The electron mass now has an explanation and a predicted value. Theorists are now challenged to evaluate the new electron equations and verify that they are consistant with all known electron properties. The gauntlet has been thrown.

Don Stevens

Repeat your calculations using Abdus Salams Yukawa micro-gravity with massive graviton such that G* = 10^40GNewton at the fermi scale.

I expect the electron evaluation equations are repeatable at the fermi scale provided that significant relationships are preserved as noted.

1. Particles will have light velocity (reference frame) spin. An observer, with a remote, fixed (constant) separation distance from an electron will see only the static electric effects of its charge and so the charge must appear to be stationary. Magnetic moment and centrifugal force result from electron reference frame spin.
2. Particles are gravitationally confined with gravitational force equal to c^4/3G. This is less than the Planck force, c^4/G newton.
3. The mass energy of a particle will be: force times radius, where radius is 3Gm/c^2 and force is c^4/3G.
E = (c^4/3G) (3Gm/c^2) = mc^2
4. The maximum energy photon wavelength (limit value) will have the energy needed to produce two particles with the mass value shown.
mcr = h/4pi
r = (h/4pi) (1/mc) = 3Gm/c^2
(h/4pi mc) (c^2/3G) = m
(h/4pi c) (c^2/3G) = m^2
(hc/12pi G)^1/2 = m
This mass value is (1/2) (2/3)^1/2 (Planck mass). The maximum energy photon wavelength is (3pi hG/c^3)^1/2.
5. For the stable electron, gravitational collapse is halted at the electron mass radius, 3Gm/c^2, where centrifugal and gravitational forces are balanced.
6. A very unique property of the maximum energy photon wavelength is its energy value. This photon energy is correctly specified by either the gravitational constant G or the Planck constant h as shown.
E = hf = h (c/wavelength)
E = (wavelength) (c^4/3pi G)
(wavelength)^2 = (hc) (3pi G/c^4)
wavelength = (3pi hG/c^3)^1/2 meter
E = (2/3)^1/2 (Planck mass) c^2
E = (hc/3pi G)^1/2 c^2 joule
7. The Einstein book, Relativity (1920) includes a footnote in Chapter 16 (page 49) that reads, "The general theory of relativity renders it likely that the electrical masses of an electron are held together by gravitational forces."
8. A spin-off from this evaluation, relates to the mass values of the electron, muon, tau, proton and neutron particles. A maximum particle mass value is defined as (hc/12pi G)^1/2. This mass, labeled (M max) is 8.88695667x10^-9 kg. A minimum particle mass value is defined as [h/(2pi)^2] (1/2c^2). This mass, labeled (M min) is 9.337376123x10^-53 kg. The ratio (M min) divided by electron mass is 1.025028393x10^-22. The ratio electron mass divided by (M max) is also 1.025028393x10^-22. The (M min) divided by (M max) is 1.050683206x10^-44. This ratio, (M min)/(M max) is a constant that is labeled (K1). The following equation applies to the mass values of electron, muon, tau, proton and neutron.
K1 = [(M min)/particle mass] [particle mass /(M max)]
For an example, the muon has the mass value, 1.883531475x10^-28 kg. With this mass, we see that (K1) is 1.050683206x10^-44. At the limit, where the particle mass is equal to (M max) then the (K1) value must be 9.337376123x10^-53 divided by (hc/12pi G)^1/2. This ratio will be 1.050683206x10^-44 only when the G value is 6.671745178x10^-11.
I have found (today 11/13/2013) that the alpha particle mass, 6.64465675x10^-27 kg, the pion pi + mass, 2.488595933x10^-28 kg and the pion pi 0 mass, 2.406593489x10^-28 kg will all obey the K1 equation where the K1 value is 1.050683206x10^-44. All of the mass values listed have a specific relationship to the (M min) and (M max) values.

I will answer the question: How do we know that the electron mass equation is not just the result of coincidence, so that it has no physical significance?

mass = (h/4pi c)(c/3pi hG)^1/4 kg
mass = (1/3G)(Le/4pi)^3 (1/2pi)^2 kg
mass = (hc/12pi G)^1/2 (Le/4pi)[1/(2pi seconds) c] kg

Answer: If one electron mass equation was the only coincidence to be explained then we might expect this could be just a coincidence. When additional equations are developed, the coincidence explanation is not credible. The following equation defines the electron Compton wavelength (labeled Le) from the electron mass and the derived G value.

Le = Compton wavelength = 2(3Gm)^1/3 (2pi)^5/3 meter
(Le/4pi)^3 = 3Gm (2pi)^2
(Le/4pi)^3 (1/3m) (1/2pi)^2 = G
G = 6.6717452x10^-11
Le = Compton wavelength = 2.4263102389x10^-12 = h/mc

Then the value h is defined as shown.

h = 2(mc)(3Gm)^1/3 (2pi)^5/3 = 6.62606957x10^-34

From the electron mass, light velocity and the derived G value, the Planck constant value is precisely defined. The coincidence explanation for these equations becomes ever less credible with each applicable equation that is developed. The known Planck constant value (h) is shown to be consistant with the derived value for the gravitational constant (G).

The electron is not smaller than 3Gm/c^2 because centrifugal force (from referenceframe spin) prevents any further collapse. The electron is not larger than 3Gm/c^2 because a larger electron radius would not have enough energy density to produce a gravitationally confined entity (mass particle).

A prophetic statement by physicist Richard Wolfson follows.

"Our current understanding of physics suggests that once an object has been squeezed to black hole size, there's no force in the Universe that can prevent its further collapse to a single point of infinite density. This infinite conclusion may change somewhat when we finally learn how to merge general relativity with quantum physics, ---. Even so, black holes will remain objects in which matter is compressed to a near point of incredible density."

This is from his book, Simply Einstein, (2003) page 218. Spin prevents the electron from collapsing "to a single point of infinite density". The electron ring singularity is "naked" as proposed by Alexander Burinskii.

Kenneth W. Ford asks the right question in his book "The Quantum World" (2004) page 23:
"Are space and time drastically warped right at the particle's (electron's) location?"
The electron requires sufficient space and time warp to achieve a stable state of confined mass energy. This stable state requires a specific quantized mass value as shown.

The Planck length is expected to play a role in a new theory of quantum gravity. As Planck observed, there is only one way to combine the constants h, G and c to obtain a distance (length). In the 1900 time frame (when Planck defined a fundamental length) we did not know that the electron had angular momentum (spin). This was learned in 1925-1927. We did not know that a photon, when converted to mass, must materialize not only an electron, but also a positive (unit) charge particle as well. The existance of the positron (antimatter) was learned in 1932. Max Planck was correct; there is only one way to combine the constants h, G and c to obtain a distance, however we must now update the fundamental length from (hG/2pi c^3)exponent 1/2 to the value (3/2)^1/2 (Planck length). This change does not alter the original Planck dimensional analysis but will allow us to relate a fundamental length value to a value that can be measured in a laboratory. We can relate the fundamental length to the electron Compton wavelength and include the electron spin property. One electron and one positron each with unit charge and 1/2 unit of angular momentum can materialize from one photon as was learned in 1932-1934.

This concept requires that we must change the basic belief that "There is no distance smaller than the Planck length that has meaning". The electron radius, (3Gm/ c squared) or 2.0287x10 exponent -57 meters is clearly smaller than the Planck length, 1.6159x10 exponent -35 meters. And yet the electron centrifugal force and gravitational force are found to be exactly balanced at the radius (3Gm/ c squared). A belief change is possible but not easily accomplished.

An equation that is undeniably correct follows.

[4pi(3Gm/c^2)/(h/mc)]^1/2 (hc/12pi G)^1/2 = electron mass
[4pi(3Gm/c^2)(mc/h)]^1/2 = electron mass/(hc/12pi G)^1/2

The derived G value allows the mass ratio to be defined as shown.

[4pi(3Gm/c^2)(mc/h]^1/2 = 1.025028393x10^-22 meter/meter
electron mass = (hc/12pi G)^1/2 (1.025028393x10^-22)
electron mass = 9.10938291x10^-31 kg

And so we must change the belief that "There is no distance smaller than the Planck length that has meaning". The electron radius, (3Gm/c^2) meter clearly has significant meaning.

In his book, The Lightness Of Being, Frank Wilczek writes (page 200) "And yet we have no good idea about why electrons weigh what they do.---We need some new ideas. At present, the best we can do is to accommodate the electron's mass as a parameter in our equations--a parameter we can't express in terms of anything more basic."

Some new ideas are presented here. We need to verify all equations used. Can we express the electron mass in terms of a fundamental mass (Planck mass) and a derived gravitational constant value? I believe the answer is: Yes we can.

When the electron mass is (h/4pi c)(c/3pi hG)^1/4 the electron mass squared is (h/4pi c)^2 (c/3pi hG)^1/2. Then the ratio m squared divided by the mass (hc/12pi G)^1/2 is useful because this allows the G value to cancel.
(m^2)/(hc/12pi G)^1/2 = (h/4pi c)^2 (c/3pi hG)^1/2 /(hc/12pi G)^1/2
(m^2)/(hc/12pi G)^1/2 = (h/4pi c) (1/2pi c)
m/(hc/12pi G)^1/2 = (h/4pi c) (1/2pi c)/ m
m = (1/m) (h/4pi c) (1/2pi c) (hc/12pi G)^1/2
m = 1.025028393x10^-22 (hc/12pi G)^1/2
m = 9.10938291x10^-31 kg

This last equation will correctly define the electron mass only when the G value is 6.671745178x10^-11. When electron mass is 9.10938291x10^-31 kg then the G value is required to be 6.671745178x10^-11 as determined earlier. We now have five separate methods that can be used to predict the electron mass value that are based on a gravitational constant value. The probability that all of these methods are incorrect is very small. We find that it is now more than, "-- likely that the electrical masses of an electron are held together by gravitational forces" (Einstein 1920).

Steven Weinberg has said, "--all logical arguments can be defeated by the simple refusal to reason logically". Readers are encouraged to verify the equations presented here and to provide feedback of findings to the writer in this Discussion Forum.

What about using regular QM Shrodinger equation as a gravitational field insted of probability field?