I will respond to a request to show a comparison of the electron pr...
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=2pi(Planck length)(3/2)^1/2=(3pi hG/c^3)^1/2
L2=1/2(electron Compton wavelength)= 1/2(Le)
L3=(1/8pi)(2/3)^1/2 (Le)^2 (1/Planck length)
L4=2pi(3Gm/c^2), where the (m) value is the electron mass
The L3 value is found from the equation:
(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 6.67174557x10^-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 person with a calculator 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 6.67174557x10^-11. No further evidence is needed to know these relationships exist.
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 1.02502838x10 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 = (1.02502838x10^-22)^2 sec/sec
Limit time factor = 1.05068319x10^-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 black hole, 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.
G = (Le/4pi)^3 (1/2pi)^2 (1/3m)
G = 6.671745573x10^-11
From the electron angular momentum, electron mass and the derived gravitational constant, a 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 = 6.626068964x10^-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 8.886956x10 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, 6.67174557x10 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.