DonJStevens

A particle that is based on the photon orbit radius (3Gm/c^2) of the particle rather than the Schwarzschild radius of the particle can be shown to have a specific relationship to the electron mass. This fundamental particle mass is defined as shown.

(mass)(velocity)(radius) = angular momentum = h/4pi

(m) c (3Gm/c^2) = h/4pi

(m)^2 = (h/4pi) (c/3G) = (hc/12pi G)

m = (hc/12pi G)^1/2 kg

This mass value is (1/2)(2/3)^1/2 (Planck mass).

m = (1/2)(2/3)^1/2 (hc/2pi G)^1/2

m = (hc/12pi G)^1/2 kg

The photon wavelength that has the correct energy to materialize a pair of particles, each with mass, (hc/12pi G)^1/2 is shown below.

h (frequency) = hc/wavelength = 2 c^2 (hc/12pi G)^1/2 J

hc/wavelength = (hc^5/3pi G)^1/2 J

(wavelength)/(hc) = (3pi G/hc^5)^1/2

wavelength = (3pi hG/c^3)^1/2 meter

wavelength = 2pi (3/2)^1/2 (Planck length) meter

The electron mass divided by the fundamental mass value (hc/12pi G)^1/2 relates to a gravitational blueshift and time dilation factor. The photon with just enough energy to materialize one electron and one positron must be gravitationally blue shifted to the wavelength (3pi hG/c^3)^1/2 in order to have the energy density needed for gravitational confinement of two particles.

wavelength / wavelength = blueshift factor meter/meter

(3pi hG/c^3)^1/2 / (h/2mc) = 1.025x10^-22 meter/meter

This blueshift factor is approximate because the gravitational constant has much greater uncertainty than the values (m) and (h). The ratio, electron mass divided by the fundamental mass value is shown below.

(electron mass)/(hc/12pi G)^1/2 = 1.025x10^-22 kg/kg

This ratio is also approximate due to uncertainty. A similar ratio with less uncertainty is:

[h/(2pi)^2] / (2mc^2) = 1.025028393x10^-22 J/J

When the two dimensionless ratio values are precisely equal, then the G value used to define the fundamental mass value must be very close to 6.671745197x10^-11. Then the electron mass divided by the fundamental mass value will be:

(electron mass)/(hc/12pi G)^1/2 = 1.025028393x10^-22 kg/kg

The electron mass is shown to be, either precisely related to the fundamental mass, (hc/12pi G)^1/2 or very closely related to this mass.

The known electron angular momentum is so high that a ring velocity greater than c would be required if the electron radius is less than 3Gm/c^2. This would not be consistent with other known electron properties. The electron can't collapse to its Schwarzschild radius because spin acceleration force and (self) gravitational force will exactly balance at the radius 3Gm/c^2. The electron can have only one (quantized) mass value because the two opposing forces are required to be balanced.

In the following equation, the value (h/mc) is the electron Compton wavelength. The value 4pi(3Gm/c^2)/(h/mc) is a dimensionless ratio meter per meter. Two equal factors of gravitational time dilation and gravitational length contraction combine to produce this meter per meter ratio. The square root of this ratio defines time dilation which has the effect of reducing the observable mass value from (hc/12pi G)^1/2 kg to the electron mass value.

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

The reader will note that this equation is exact. However we must know the true G value if we are to know the numerical ratio of electron mass to the fundamental mass value. This mass equation defines a clear relationship between the fundamental mass, (hc/12pi G)^1/2 kg and the electron mass.

We find the fundamental mass, (hc/12pi G)^1/2 kg, the electron mass and the electron Compton wavelength numbers (values) fall neatly in place when the G value, 6.671745197x10^-11 is used. A limit time dilation factor labeled (Tf) is defined next as shown. One second is the time required for light to travel 299792458 meters.

Tf = (3/2)^1/2 (Planck time)/(2pi seconds)

Tf = (3/2)^1/2 (hG/2pi c^5)^1/2 / (2pi seconds)

Tf = 1.050683207x10^-44 second/second

(Tf)^1/2 = 1.025028393x10^-22 second/second

(Tf)^1/2 = (electron mass)/(hc/12pi G)^1/2 kg/kg

electron mass = (hc/12pi G)^1/2 (Tf)^1/2 kg

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

This electron mass equation is approximately correct when any recently published value for the gravitational constant G is used. The photon wavelength, (h/2mc) with the energy to produce one electron plus one positron, is related to the short wavelength, (3pi hG/c^3)^1/2 meter as shown.

(h/2mc)(Tf)^1/2 = (3pi hG/c^3)^1/2 meter

(h/2mc) = (3pi hG/c^3)^1/2 / (Tf)^1/2 meter

electron Compton wavelength = (h/mc) meter

(h/mc) = 2(3pi hG/c^3)^1/2 / (Tf)^1/2 meter

(h/mc) = 4pi(3pi hG/c)^1/4 meter

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

(h/4pi mc)^4 = (3pi hG/c)

(h/4pi mc)^4 (c/3pi h) = G = 6.671745178x10^-11

We now see that the electron mass and the electron Compton wavelength are correctly specified by using the well known Planck constant, the velocity of light and the constant 6.6717452x10^-11. This third constant is most probably, a more precise value for the gravitational constant. Measurements performed to determine the gravitational constant value are difficult and have large uncertainty and so this constant is found to be the gravitational constant with reduced uncertainty.

In the following equation, the (Le) value is the electron Compton wavelength.

electron m = (hc/12pi G)^1/2 (Le/4pi)[1/(2pi c)(one sec)]kg

(Le/4pi)[1/(2pi c)(one sec)] = 1.025028394x10^-22 = (Tf)^1/2 sec/sec

(Le/4pi)^2 (1/2pi c)^2 = Tf = 1.050683207x10^-44 sec/sec

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

(Le/4pi)^2 (1/2pi c)^2 = (3hG/4pi c^5)^1/2 /(2pi seconds)

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

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

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

(Le/4pi)^4 (c/3pi h) = G = 6.671745178x10^-11 N m^2/kg^2

(3pi hG/c)^1/2 (1/2pi c)^2 = (3hG/4pi c^5)^1/2 /(2pi seconds)

(3pi hG/c)^1/2 (1/2pi c)^2 = Tf sec/sec

Fundamental constants now define the quantized electron mass value and specify its relationship to the Planck mass. The electron is found to be a gravitationally confined entity that is gravitationally collapsed to its radius 3Gm/c^2 where forces are balanced. It is not collapsed to infinite density and so it is not collapsed to a point.

Any spinning mass that is gravitationally collapsed, will have two specific radius values that are defined by the mass. There is the "static limit" radius, and also the "photon orbit" radius. When angular momentum is high, the photon orbit radius is smaller than the static limit radius. Within the static limit, the reference frame must spin. At the photon orbit radius, the reference frame spins at light velocity. The zero mass electric field that we refer to as a fundamental charge will appear to be stationary (from a distance) because its reference frame spins at light velocity. Only when observer separation distance is changing, will the electron be observed to rotate (spin). With light velocity reference frame spin, the time rate within the radius, 3Gm/ c squared is zero seconds per second. The self gravitational field or space curvature reaches its limit at the electron mass photon orbit radius. See "Qualitative representation of horizon, ergosphere, and static limit" page 880 of book, Gravitation by Misner, Thorne and Wheeler (1970-1971).

From a derived value for the gravitational constant, the Planck constant value is specified as shown.

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

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

The CODATA value for h is 6.62606957x10 exponent -34. The electron mass, light velocity and the gravitational constant determine the Planck constant value. Confirmation that either of the two equations shown above is correct, will confirm that the quantized electron mass equation is correct. Electron mass is equal to (h/4pi c) times (c/3pi hG) exponent 1/4. The equation that defines the h value uses only fundamental constants, electron mass, light velocity and the gravitational constant G. There is only one G value that will satisfy this equation. This G value is then identified as a fundamental constant.

The CODATA value for h bar is 1.054571726x10^-34.

h bar = h/2pi = 2(mc)(3Gm)^1/3 (2pi)^2/3

The only G value that will precisely satisfy this h bar equation is 6.671745178x10^-11. The tiny difference in the two G values is not significant because we have a small uncertainty in the electron mass and h bar values.

In 1933 a photograph was taken by Irene Joliot-Curie and Fredric Joliot showing the conversion of a single photon into two mass particles. This provided early evidence that electromagnetic energy could be converted to a pair of mass particles.

A photon has equal amounts of positive and negative electric field energy so the conversion of a photon into two particles, each with equal positive and negative charge is consistent. Quantized photon energy increases as frequency increases. Photon energy density increases as the photon wavelength becomes smaller.

Theorists have expected that an energy density limit will be found when wavelength becomes very small and energy density is large. Max Planck defined a limit condition when he defined Planck mass and Planck energy. The Planck mass value implies infinite energy density and implies that a photon could be converted to a single Planck mass particle. The "single" particle from photon energy and the infinite mass density have never been adequately explained. A very important accomplishment by Planck was the finding from dimensional analysis, that a specific mass value, (hc/2pi G) exponent 1/2 can be specified using three fundamental constants.

A photon with less than the Planck energy can be converted to a particle pair without requiring unrealistic infinite energy density. The original Planck dimensional analysis applies and so the units of mass, length and time are consistent. This photon with slightly less energy than the Planck energy has the wavelength 2pi (3/2) exponent 1/2 (Planck length). This value is (3pi hG/c^3) exponent 1/2 meters. The value (3/2) exponent 1/2 times (Planck length) is a radius that determines charge acceleration.

radius = (3/2)^1/2 (hG/2pi c^3)^1/2 meter

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

When the derived G value, 6.671745196x10 exp -11 is used and 6.62606957x10 exp -34 is used as the Planck constant, the electron mass has the value shown.

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

mass = 9.109382904x10^-31 kg

When the derived G value, 6.671745178x10 exp -11 is used and 6.62606957x10 exp -34 is used as the Planck constant, the electron mass has the value shown.

mass = 9.10938291x10^-31 kg

The CODATA electron mass is 9.10938291x10 exp -31 kg with Standard uncertainty 0.000 000 40 x10 exp -31 kg.

The fine structure constant labeled (alpha) is related to the von Klitzing constant labeled (Rk) as shown.

1/alpha = (h/e^2) (2/Zo) = 137.0359991

(h/e^2) = (Zo/2) (137.0359991)

(h/e^2) = (376.7303135) (1/2) (137.0359991)

(h/e^2) = 25812.80745 ohm = Rk

h/(2e)^2 = 6453.201863 ohm = Rk/4

The minimum alternating current required to radiate one photon per cycle will be two charges per cycle. Two energy pulses will be present during each cycle. This current is (2e) times frequency. The value labeled (Kj) is the Josephson constant. The value 1/(Kj) is (h/2e).

(2e) (frequency) [h/(2e)^2] = (current) (impedance)

(2e) (frequency) [h/(2e)^2] = voltage

(frequency) (h/2e) = (frequency) [1/(Kj)] = voltage

When wavelength is (h/2mc) meter then frequency is (2mc^2/h) Hz.

(2mc^2/h) (h/2e) = (mc^2)/e = 510998.9276 volts

capacitance = (e^2)/ (mc^2) = 3.135381463x10^-25 farad

When impedance is 6453.201863 ohm or (Rk/4) then inductance has the value shown.

(Rk/4)^2 (3.135381463x10^-25 farad) = inductance

electron inductance = 1.305692434x10^-17 H

The electron charge energy after gravitational collapse to the radius (3Gm/c^2) labeled R4, is force times radius.

R4 = 3Gm/c^2 meter

m = (R4) (c^2/3G) = (3Gm/c^2) (c^2/3G) kg

mc^2 = (3Gm/c^2) (c^4/3G) J

E = (3Gm/c^2) (c^4/3G) J

Prior to gravitational collapse, when the radius is far larger, the force value (c^4/3G) newton must be proportionally smaller in order to define energy. The larger radius labeled R2 is (h/4pi mc) meter. This radius is the photon wavelength (h/2mc) meter divided by 2pi.

R2 = h/4pi mc meter

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

R2/R4 = (hc/12pi m^2 G) meter/meter

R4/R2 = (12pi m^2 G/hc) meter/meter

R4/R2 = 1.050683209x10^-44 meter/meter

For the larger radius R2, force is reduced by the factor R4/R2.

force = (c^4/3G) (12pi m^2 G/hc) newton

E = (c^4/3G) (12pi m^2 G/hc) (R2) J

E = (c^4/3G) (12pi m^2 G/hc) (h/4pi mc) J

E = mc^2 J

The photon wavelength,(h/2mc) meter has energy equal to 2E or 2mc squared joule. This photon can produce two mass particles as observed by Irene Joliot-Curie and Fredric Joliot, 1933.

The ratio R4 divided by R2 has the units meter per meter. This dimensionless ratio defines the product of gravitational length crontraction and gravitational time dilation. The dimensionless length contraction factor will have the value (R4/R2) exponent 1/2 while the time dilation factor will also have the value (R4/R2) exponent 1/2.

(R4/R2)^1/2 = (Tf)^1/2

R4/R2 = (3/2)^1/2 (Planck time)/2pi seconds

R4/R2 = (12pi m^2 G/hc) meter/meter

(12pi m^2 G/hc) = (3/2)^1/2 (hG/2pi c^5)^1/2 /(2pi)

In the equation above, a dimensionless ratio of length divided by length is equal to a dimensionless ratio of seconds divided by seconds. The m value is electron mass as noted earlier. This is consistent because the gravitational time dilation factor is known to be equal to the gravitational length contraction factor. The dimensionless ratio equation can be solved for a G value so that the G value is required to be 6.671745178x10 exponent -11 N m^2/kg^2. We now have strong evidence confirming the electron mass equation.

G^1/2 = [h/24 (pi)^2 (m)^2 c] (3h/4pi c)^1/2

G = 6.671745178x10^-11 N m^2/kg^2

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

This is the fundamental mass value (hc/12pi G)^1/2 kg after being degraded (reduced) by the dimensionless time dilation factor (R4/R2) exponent 1/2. From the electron mass equation, a gravitational constant value is derived.

G(m)^4 = (h/4pi c)^4 (c/3pi h)

G = (h/4pi c m)^4 (c/3pi h)

G = 6.671745178x10^-11 N m^2/kg^2

(R4/R2)^1/2 = 1.025028393x10^-22 meter/meter

(R4/R2)^1/2 (hc/12pi G)^1/2 = 9.10938291x10^-31 kg

The electron mass is also defined by each of the two equations shown below. The c value is one light second or 2.99792458 meters.

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

mass = (hc/12pi G)^1/2 (Le/4pi)(1/2pi c) kg

In each electron mass equation, the constant G is required to have the value 6.671745178x10^-11 N m^2/kg^2. This is found to be a fundamental constant value.

We will now direct our attention to the proton mass. This mass value is 1.672621777x10 exp -27 kg. The proton mass divided by the fundamental mass, (hc/12pi G)^1/2 is equal to the square root of the ratio, 4pi(3Gm/c^2) divided by the proton Compton wavelength. The proton Compton wavelength is (h/mc).

[4pi(3Gm/c^2)/(h/mc)]^1/2 = 1.882108622x10^-19

The fundamental mass value, (hc/12pi G)^1/2 multiplied by the dimensionless factor 1.882108622x10^-19 is equal to the proton mass value 1.672621777x10^-27 kg. The electron and the proton are now shown to have a specific relationship to the Planck mass. The proton mass equation provides one more opportunity to verify the gravitational constant value G.

proton mass = (hc/12pi G)^1/2 (1.882108622x10^-19)

G^1/2 (proton mass) = (1.882108622x10^-19)(hc/12pi)^1/2

G^1/2 = (1/proton mass)(1.882108622x10^-19)(hc/12pi)^1/2

G = 6.671745177x10^-11 N m^2/kg^2

The logic sequence that is the basis for this derived G value is very specific and easy to verify.

With evidence showing the derived G value is related to electron mass, we can now write the equation for a photon wavelength that has energy specified by either the Planck constant or the gravitational constant. This short wavelength photon has maximum (limit) energy. The photon energy value is (2/3) exponent 1/2 times the Planck energy. This wavelength meets the Planck requirement where photon energy is h times frequency and also meets a limit requirement where photon energy is wavelength times (c^4/3pi G).

E = h (frequency) = h (c/wavelength)

E = wavelength (c^4/3pi G)

h (c/wavelength) = wavelength (c^4/3pi G)

(hc)(3pi G/c^4) = (wavelength)^2

wavelength = (3pi hG/c^3)^1/2 m

Readers will note that this wavelength is 2pi (3/2)^1/2 (Planck length). This wavelength photon has energy equal to (2/3)^1/2 times (Planck energy). The wavelength of any photon determines its energy and determines how its energy relates to the Planck energy.

Limit frequency = c/wavelength = c/(3pi hG/c^3)^1/2

Limit frequency = 2.410840467x10^42 Hz

E = (2/3)^1/2 (Planck mass) c^2 J

E = (2/3)^1/2 (hc/2pi G)^1/2 c^2 J

E = (hc/3pi G)^1/2 c^2 J

E = (hc^5/3pi G)^1/2 J

When photon frequency is known or photon wavelength is known, the photon energy is determined as shown.

(photon frequency/limit frequency) (hc^5/3pi G)^1/2 = E

When frequency is c/wavelength or c/ 1.213155119x10^-12 meter, frequency is c(2mc/h) or 2.471179929x10^20 Hz.

(2.471179929x10^20 Hz)/(2.410840467x10^42 Hz) = 1.025028393x10^-22

This is the same dimensionless ratio that has been previously found. This ratio is:

(R4/R2)^1/2 = (Tf)^1/2 = 1.025028393x10^-22

1.025028393x10^-22 (hc^5/3pi G)^1/2 = 2(electron mass)c^2 J

1.025028393x10^-22 (hc^5/3pi G)^1/2 = 1.637421013x10^-13 J

We can show that the maximum photon energy value (hc^5/3pi G)^1/2 J is reduced to h/(2pi)^2 J when subjected to the Tf time dilation factor.

Tf = (3hG/4pi c^5)^1/2 second / 2pi second

(hc^5/3pi G)^1/2 (Tf) = h/(2pi)^2 J

(hc^5/3pi G)^1/2 (3hG/4pi c^5)^1/2 (1/2pi) = h/(2pi)^2 J

This is an exact solution because the gravitational constant is canceled. Two energy values are precisely specified. These are labeled E2 and E3. The m value is electron mass.

E2 = 2mc^2 J

E3 = h/(2pi)^2 J

The E1 value is hypothesized to be the maximum photon energy. These energy values are related as shown. The E1 value is (hc^5/3pi G)^1/2 joule.

E1/E2 = E2/E3

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

4 m^2 c^4 = (hc^5/3pi G)^1/2 [h/(2pi)^2]

(G)^1/2 = (hc^5/3pi)^1/2 [h/(2pi)^2] (1/4 m^2 c^4)

G = 6.671745178x10^-11 N m^2/kg^2

This G value is shown to be consistent with the electron mass equations. This G value is closely related to the Planck constant.

In this concept, the gravitational collapse of a photon with wavelength (h/2mc) meters requires two phases. The first phase increases photon energy density. There is no change in photon energy because time dilation is matched to wavelength shortening. The photon wavelength (h/2mc) meter is shortened to the limit wavelength in this phase. The limit wavelength is (3pi hG/c^3) exponent 1/2 meter.

(3pi hG/c^3)^1/2 / (h/2mc) = 1.025028393x10^-22 m/m

While wavelength is reduced, time rate is reduced, so that available photon energy is constant. The next phase is the conversion of electromagnetic energy into gravitational field energy. In this phase two mass particles are materialized. The two particle reference frames spin at light velocity. The final resulting time rate at each particle radius (3Gm/c^2) will then approach zero seconds per second.

When an electron and a positron merge together, their reference frame angular momentum values are added. These opposite values will sum to zero. Each collapsed particle will have zero temperature only when spin is maximal. When angular momentum is canceled, the particles immediately become hot and radiate their energy away. The energy emitted in radiation is compensated for by a decrease in the mass of the particle pair.

Leonard Susskind has described a limitation on the smallness of things as a cutoff. A quote from Susskind follows.

"A cutoff sounds like a copout. Physicists have long speculated that the Planck length is the ultimate atom of space. Feynman diagrams, even those involving gravitons, make perfect sense as long as you cease adding structures smaller than the Planck length - or so the argument goes. This was the almost universal expectation about space-time- that it would have an indivisable, granular, voxelated structure at the Planck scale."

Evidence has now been presented showing that the limitation on the smallness of the electron is the radius 3Gm/c squared. When this limit is incorporated, the product of the Planck constant and the gravitational constant can be precisely defined.

h G = (6.62606957x10^-34) (6.671745178x10^-11)

A quote from Kenneth W. Ford follows.

"-- natural units h and c are arbitrary,-- but scientists can agree that they are directly related to fundamental features of the natural world. For an all-natural physics, we need one more natural unit, which has yet to emerge. This unit if it is found, may be a length or a time."

The natural length defined by this writer is the photon wavelength h/2mc meters, where m is electron mass.

h/2mc = 1.213155119x10^-12 m

When the derived G value is used, h/2mc is also equal to the value shown.

2pi (3pi hG/c)^1/4 = 1.213155119x10^-12 m

This wavelength uses the product of h and G to define a length value as proposed by Max Planck in 1900. A second quote from Kenneth W. Ford follows.

"Such a unit could usher in a whole new view of space and time in the subatomic world --- in layers of reality far below those explored so far."

The quotes are from the book, The Quantum World (page 28) by author Ford (2004).

Those readers who are interested in the quantized electron mass will want to read Chapter 8 from the book, Dreams Of A Final Theory, by Steven Weinberg (1992). Two significant statements are quoted below.

"--we think that all the forces of nature (including gravity) become united at something like the Planck energy,----." (page 206)

"But, because the standard model leaves out gravitation, we now think that it is only a low-energy approximation to a really fundamental unified theory and that it loses its validity at energies like the Planck energy." (page 207)

We now have evidence showing that electromagnetic and gravitational forces become united (equal) at the photon wavelength (3pi hG/c^3)^1/2 meter. A photon with this wavelength has the energy value (2/3)^1/2 times the Planck energy. This is the energy value labeled E1.

E1 = (2/3)^1/2 (hc^5/2pi G)^1/2 = (hc^5/3pi G)^1/2

The mass energy of the electron relates to two energy values. One energy value, labeled E4 is very small while the other energy value, labeled E5 is much larger.

E4 = (h/2)(1/2pi)^2 joule

E5 = (hc/12pi G)^1/2 (c squared) joule

(E4)^1/2 (E5)^1/2 = 8.187105066X10^-14 joule

(E4)^1/2 (E5)^1/2 (1/c squared) = 9.10938291x10^-31kg

The applicable value for the gravitational constant, G

is defined from the equation shown below.

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

G = 6.671745178x10^-11 N m^2/kg^2

This G value was previously determined from a number of separate evaluations. The dimensionless ratio, E4 divided by E5 has the value 1.050683206x10^-44 joule/joule. The same dimensionless ratio has the value seconds/second as shown below.

(3/2)^1/2 (Planck time)/(2pi seconds) = 1.050683206x10^-44 = seconds/second

I have described this ratio as a dimensionless time dilation limit. This ratio or the square root of this ratio shows up repeatedly in electron mass evaluations as shown below.

(1.050683206x10^-44)^1/2 (hc/12pi G)^1/2 = electron mass

The equation that defines a photon wavelength with energy specified by either the Planck constant or the gravitational constant is a significant part of this concept.

Energy = h f = h (c/wavelength)

Energy = (wavelength) (c^4/3pi G)

h (c/wavelength) = (wavelength) (c^4/3pi G)

(wavelength)^2 = (hc) (3pi G/c^4)

wavelength = (3pi hG/c^3)^1/2 meter

Energy = (2/3)^1/2 (Planck mass) c^2

Energy = (hc/ 3pi G)^1/2 c^2 joule

This upper energy limit provides a new interpretation of the Planck constant. Any photon wavelength will have energy that is equal to, or less than (2/3) exponent 1/2 times (Planck mass) times c squared. At any greater energy, the equation, E = h (frequency) does not apply.

Readers are asked to provide feedback to this writer if any error is found in equations presented here.

(mass)(velocity)(radius) = angular momentum = h/4pi

(m) c (3Gm/c^2) = h/4pi

(m)^2 = (h/4pi) (c/3G) = (hc/12pi G)

m = (hc/12pi G)^1/2 kg

This mass value is (1/2)(2/3)^1/2 (Planck mass).

m = (1/2)(2/3)^1/2 (hc/2pi G)^1/2

m = (hc/12pi G)^1/2 kg

The photon wavelength that has the correct energy to materialize a pair of particles, each with mass, (hc/12pi G)^1/2 is shown below.

h (frequency) = hc/wavelength = 2 c^2 (hc/12pi G)^1/2 J

hc/wavelength = (hc^5/3pi G)^1/2 J

(wavelength)/(hc) = (3pi G/hc^5)^1/2

wavelength = (3pi hG/c^3)^1/2 meter

wavelength = 2pi (3/2)^1/2 (Planck length) meter

The electron mass divided by the fundamental mass value (hc/12pi G)^1/2 relates to a gravitational blueshift and time dilation factor. The photon with just enough energy to materialize one electron and one positron must be gravitationally blue shifted to the wavelength (3pi hG/c^3)^1/2 in order to have the energy density needed for gravitational confinement of two particles.

wavelength / wavelength = blueshift factor meter/meter

(3pi hG/c^3)^1/2 / (h/2mc) = 1.025x10^-22 meter/meter

This blueshift factor is approximate because the gravitational constant has much greater uncertainty than the values (m) and (h). The ratio, electron mass divided by the fundamental mass value is shown below.

(electron mass)/(hc/12pi G)^1/2 = 1.025x10^-22 kg/kg

This ratio is also approximate due to uncertainty. A similar ratio with less uncertainty is:

[h/(2pi)^2] / (2mc^2) = 1.025028393x10^-22 J/J

When the two dimensionless ratio values are precisely equal, then the G value used to define the fundamental mass value must be very close to 6.671745197x10^-11. Then the electron mass divided by the fundamental mass value will be:

(electron mass)/(hc/12pi G)^1/2 = 1.025028393x10^-22 kg/kg

The electron mass is shown to be, either precisely related to the fundamental mass, (hc/12pi G)^1/2 or very closely related to this mass.

The known electron angular momentum is so high that a ring velocity greater than c would be required if the electron radius is less than 3Gm/c^2. This would not be consistent with other known electron properties. The electron can't collapse to its Schwarzschild radius because spin acceleration force and (self) gravitational force will exactly balance at the radius 3Gm/c^2. The electron can have only one (quantized) mass value because the two opposing forces are required to be balanced.

In the following equation, the value (h/mc) is the electron Compton wavelength. The value 4pi(3Gm/c^2)/(h/mc) is a dimensionless ratio meter per meter. Two equal factors of gravitational time dilation and gravitational length contraction combine to produce this meter per meter ratio. The square root of this ratio defines time dilation which has the effect of reducing the observable mass value from (hc/12pi G)^1/2 kg to the electron mass value.

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

The reader will note that this equation is exact. However we must know the true G value if we are to know the numerical ratio of electron mass to the fundamental mass value. This mass equation defines a clear relationship between the fundamental mass, (hc/12pi G)^1/2 kg and the electron mass.

We find the fundamental mass, (hc/12pi G)^1/2 kg, the electron mass and the electron Compton wavelength numbers (values) fall neatly in place when the G value, 6.671745197x10^-11 is used. A limit time dilation factor labeled (Tf) is defined next as shown. One second is the time required for light to travel 299792458 meters.

Tf = (3/2)^1/2 (Planck time)/(2pi seconds)

Tf = (3/2)^1/2 (hG/2pi c^5)^1/2 / (2pi seconds)

Tf = 1.050683207x10^-44 second/second

(Tf)^1/2 = 1.025028393x10^-22 second/second

(Tf)^1/2 = (electron mass)/(hc/12pi G)^1/2 kg/kg

electron mass = (hc/12pi G)^1/2 (Tf)^1/2 kg

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

This electron mass equation is approximately correct when any recently published value for the gravitational constant G is used. The photon wavelength, (h/2mc) with the energy to produce one electron plus one positron, is related to the short wavelength, (3pi hG/c^3)^1/2 meter as shown.

(h/2mc)(Tf)^1/2 = (3pi hG/c^3)^1/2 meter

(h/2mc) = (3pi hG/c^3)^1/2 / (Tf)^1/2 meter

electron Compton wavelength = (h/mc) meter

(h/mc) = 2(3pi hG/c^3)^1/2 / (Tf)^1/2 meter

(h/mc) = 4pi(3pi hG/c)^1/4 meter

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

(h/4pi mc)^4 = (3pi hG/c)

(h/4pi mc)^4 (c/3pi h) = G = 6.671745178x10^-11

We now see that the electron mass and the electron Compton wavelength are correctly specified by using the well known Planck constant, the velocity of light and the constant 6.6717452x10^-11. This third constant is most probably, a more precise value for the gravitational constant. Measurements performed to determine the gravitational constant value are difficult and have large uncertainty and so this constant is found to be the gravitational constant with reduced uncertainty.

In the following equation, the (Le) value is the electron Compton wavelength.

electron m = (hc/12pi G)^1/2 (Le/4pi)[1/(2pi c)(one sec)]kg

(Le/4pi)[1/(2pi c)(one sec)] = 1.025028394x10^-22 = (Tf)^1/2 sec/sec

(Le/4pi)^2 (1/2pi c)^2 = Tf = 1.050683207x10^-44 sec/sec

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

(Le/4pi)^2 (1/2pi c)^2 = (3hG/4pi c^5)^1/2 /(2pi seconds)

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

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

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

(Le/4pi)^4 (c/3pi h) = G = 6.671745178x10^-11 N m^2/kg^2

(3pi hG/c)^1/2 (1/2pi c)^2 = (3hG/4pi c^5)^1/2 /(2pi seconds)

(3pi hG/c)^1/2 (1/2pi c)^2 = Tf sec/sec

Fundamental constants now define the quantized electron mass value and specify its relationship to the Planck mass. The electron is found to be a gravitationally confined entity that is gravitationally collapsed to its radius 3Gm/c^2 where forces are balanced. It is not collapsed to infinite density and so it is not collapsed to a point.

Any spinning mass that is gravitationally collapsed, will have two specific radius values that are defined by the mass. There is the "static limit" radius, and also the "photon orbit" radius. When angular momentum is high, the photon orbit radius is smaller than the static limit radius. Within the static limit, the reference frame must spin. At the photon orbit radius, the reference frame spins at light velocity. The zero mass electric field that we refer to as a fundamental charge will appear to be stationary (from a distance) because its reference frame spins at light velocity. Only when observer separation distance is changing, will the electron be observed to rotate (spin). With light velocity reference frame spin, the time rate within the radius, 3Gm/ c squared is zero seconds per second. The self gravitational field or space curvature reaches its limit at the electron mass photon orbit radius. See "Qualitative representation of horizon, ergosphere, and static limit" page 880 of book, Gravitation by Misner, Thorne and Wheeler (1970-1971).

From a derived value for the gravitational constant, the Planck constant value is specified as shown.

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

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

The CODATA value for h is 6.62606957x10 exponent -34. The electron mass, light velocity and the gravitational constant determine the Planck constant value. Confirmation that either of the two equations shown above is correct, will confirm that the quantized electron mass equation is correct. Electron mass is equal to (h/4pi c) times (c/3pi hG) exponent 1/4. The equation that defines the h value uses only fundamental constants, electron mass, light velocity and the gravitational constant G. There is only one G value that will satisfy this equation. This G value is then identified as a fundamental constant.

The CODATA value for h bar is 1.054571726x10^-34.

h bar = h/2pi = 2(mc)(3Gm)^1/3 (2pi)^2/3

The only G value that will precisely satisfy this h bar equation is 6.671745178x10^-11. The tiny difference in the two G values is not significant because we have a small uncertainty in the electron mass and h bar values.

In 1933 a photograph was taken by Irene Joliot-Curie and Fredric Joliot showing the conversion of a single photon into two mass particles. This provided early evidence that electromagnetic energy could be converted to a pair of mass particles.

A photon has equal amounts of positive and negative electric field energy so the conversion of a photon into two particles, each with equal positive and negative charge is consistent. Quantized photon energy increases as frequency increases. Photon energy density increases as the photon wavelength becomes smaller.

Theorists have expected that an energy density limit will be found when wavelength becomes very small and energy density is large. Max Planck defined a limit condition when he defined Planck mass and Planck energy. The Planck mass value implies infinite energy density and implies that a photon could be converted to a single Planck mass particle. The "single" particle from photon energy and the infinite mass density have never been adequately explained. A very important accomplishment by Planck was the finding from dimensional analysis, that a specific mass value, (hc/2pi G) exponent 1/2 can be specified using three fundamental constants.

A photon with less than the Planck energy can be converted to a particle pair without requiring unrealistic infinite energy density. The original Planck dimensional analysis applies and so the units of mass, length and time are consistent. This photon with slightly less energy than the Planck energy has the wavelength 2pi (3/2) exponent 1/2 (Planck length). This value is (3pi hG/c^3) exponent 1/2 meters. The value (3/2) exponent 1/2 times (Planck length) is a radius that determines charge acceleration.

radius = (3/2)^1/2 (hG/2pi c^3)^1/2 meter

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

When the derived G value, 6.671745196x10 exp -11 is used and 6.62606957x10 exp -34 is used as the Planck constant, the electron mass has the value shown.

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

mass = 9.109382904x10^-31 kg

When the derived G value, 6.671745178x10 exp -11 is used and 6.62606957x10 exp -34 is used as the Planck constant, the electron mass has the value shown.

mass = 9.10938291x10^-31 kg

The CODATA electron mass is 9.10938291x10 exp -31 kg with Standard uncertainty 0.000 000 40 x10 exp -31 kg.

The fine structure constant labeled (alpha) is related to the von Klitzing constant labeled (Rk) as shown.

1/alpha = (h/e^2) (2/Zo) = 137.0359991

(h/e^2) = (Zo/2) (137.0359991)

(h/e^2) = (376.7303135) (1/2) (137.0359991)

(h/e^2) = 25812.80745 ohm = Rk

h/(2e)^2 = 6453.201863 ohm = Rk/4

The minimum alternating current required to radiate one photon per cycle will be two charges per cycle. Two energy pulses will be present during each cycle. This current is (2e) times frequency. The value labeled (Kj) is the Josephson constant. The value 1/(Kj) is (h/2e).

(2e) (frequency) [h/(2e)^2] = (current) (impedance)

(2e) (frequency) [h/(2e)^2] = voltage

(frequency) (h/2e) = (frequency) [1/(Kj)] = voltage

When wavelength is (h/2mc) meter then frequency is (2mc^2/h) Hz.

(2mc^2/h) (h/2e) = (mc^2)/e = 510998.9276 volts

capacitance = (e^2)/ (mc^2) = 3.135381463x10^-25 farad

When impedance is 6453.201863 ohm or (Rk/4) then inductance has the value shown.

(Rk/4)^2 (3.135381463x10^-25 farad) = inductance

electron inductance = 1.305692434x10^-17 H

The electron charge energy after gravitational collapse to the radius (3Gm/c^2) labeled R4, is force times radius.

R4 = 3Gm/c^2 meter

m = (R4) (c^2/3G) = (3Gm/c^2) (c^2/3G) kg

mc^2 = (3Gm/c^2) (c^4/3G) J

E = (3Gm/c^2) (c^4/3G) J

Prior to gravitational collapse, when the radius is far larger, the force value (c^4/3G) newton must be proportionally smaller in order to define energy. The larger radius labeled R2 is (h/4pi mc) meter. This radius is the photon wavelength (h/2mc) meter divided by 2pi.

R2 = h/4pi mc meter

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

R2/R4 = (hc/12pi m^2 G) meter/meter

R4/R2 = (12pi m^2 G/hc) meter/meter

R4/R2 = 1.050683209x10^-44 meter/meter

For the larger radius R2, force is reduced by the factor R4/R2.

force = (c^4/3G) (12pi m^2 G/hc) newton

E = (c^4/3G) (12pi m^2 G/hc) (R2) J

E = (c^4/3G) (12pi m^2 G/hc) (h/4pi mc) J

E = mc^2 J

The photon wavelength,(h/2mc) meter has energy equal to 2E or 2mc squared joule. This photon can produce two mass particles as observed by Irene Joliot-Curie and Fredric Joliot, 1933.

The ratio R4 divided by R2 has the units meter per meter. This dimensionless ratio defines the product of gravitational length crontraction and gravitational time dilation. The dimensionless length contraction factor will have the value (R4/R2) exponent 1/2 while the time dilation factor will also have the value (R4/R2) exponent 1/2.

(R4/R2)^1/2 = (Tf)^1/2

R4/R2 = (3/2)^1/2 (Planck time)/2pi seconds

R4/R2 = (12pi m^2 G/hc) meter/meter

(12pi m^2 G/hc) = (3/2)^1/2 (hG/2pi c^5)^1/2 /(2pi)

In the equation above, a dimensionless ratio of length divided by length is equal to a dimensionless ratio of seconds divided by seconds. The m value is electron mass as noted earlier. This is consistent because the gravitational time dilation factor is known to be equal to the gravitational length contraction factor. The dimensionless ratio equation can be solved for a G value so that the G value is required to be 6.671745178x10 exponent -11 N m^2/kg^2. We now have strong evidence confirming the electron mass equation.

G^1/2 = [h/24 (pi)^2 (m)^2 c] (3h/4pi c)^1/2

G = 6.671745178x10^-11 N m^2/kg^2

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

This is the fundamental mass value (hc/12pi G)^1/2 kg after being degraded (reduced) by the dimensionless time dilation factor (R4/R2) exponent 1/2. From the electron mass equation, a gravitational constant value is derived.

G(m)^4 = (h/4pi c)^4 (c/3pi h)

G = (h/4pi c m)^4 (c/3pi h)

G = 6.671745178x10^-11 N m^2/kg^2

(R4/R2)^1/2 = 1.025028393x10^-22 meter/meter

(R4/R2)^1/2 (hc/12pi G)^1/2 = 9.10938291x10^-31 kg

The electron mass is also defined by each of the two equations shown below. The c value is one light second or 2.99792458 meters.

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

mass = (hc/12pi G)^1/2 (Le/4pi)(1/2pi c) kg

In each electron mass equation, the constant G is required to have the value 6.671745178x10^-11 N m^2/kg^2. This is found to be a fundamental constant value.

We will now direct our attention to the proton mass. This mass value is 1.672621777x10 exp -27 kg. The proton mass divided by the fundamental mass, (hc/12pi G)^1/2 is equal to the square root of the ratio, 4pi(3Gm/c^2) divided by the proton Compton wavelength. The proton Compton wavelength is (h/mc).

[4pi(3Gm/c^2)/(h/mc)]^1/2 = 1.882108622x10^-19

The fundamental mass value, (hc/12pi G)^1/2 multiplied by the dimensionless factor 1.882108622x10^-19 is equal to the proton mass value 1.672621777x10^-27 kg. The electron and the proton are now shown to have a specific relationship to the Planck mass. The proton mass equation provides one more opportunity to verify the gravitational constant value G.

proton mass = (hc/12pi G)^1/2 (1.882108622x10^-19)

G^1/2 (proton mass) = (1.882108622x10^-19)(hc/12pi)^1/2

G^1/2 = (1/proton mass)(1.882108622x10^-19)(hc/12pi)^1/2

G = 6.671745177x10^-11 N m^2/kg^2

The logic sequence that is the basis for this derived G value is very specific and easy to verify.

With evidence showing the derived G value is related to electron mass, we can now write the equation for a photon wavelength that has energy specified by either the Planck constant or the gravitational constant. This short wavelength photon has maximum (limit) energy. The photon energy value is (2/3) exponent 1/2 times the Planck energy. This wavelength meets the Planck requirement where photon energy is h times frequency and also meets a limit requirement where photon energy is wavelength times (c^4/3pi G).

E = h (frequency) = h (c/wavelength)

E = wavelength (c^4/3pi G)

h (c/wavelength) = wavelength (c^4/3pi G)

(hc)(3pi G/c^4) = (wavelength)^2

wavelength = (3pi hG/c^3)^1/2 m

Readers will note that this wavelength is 2pi (3/2)^1/2 (Planck length). This wavelength photon has energy equal to (2/3)^1/2 times (Planck energy). The wavelength of any photon determines its energy and determines how its energy relates to the Planck energy.

Limit frequency = c/wavelength = c/(3pi hG/c^3)^1/2

Limit frequency = 2.410840467x10^42 Hz

E = (2/3)^1/2 (Planck mass) c^2 J

E = (2/3)^1/2 (hc/2pi G)^1/2 c^2 J

E = (hc/3pi G)^1/2 c^2 J

E = (hc^5/3pi G)^1/2 J

When photon frequency is known or photon wavelength is known, the photon energy is determined as shown.

(photon frequency/limit frequency) (hc^5/3pi G)^1/2 = E

When frequency is c/wavelength or c/ 1.213155119x10^-12 meter, frequency is c(2mc/h) or 2.471179929x10^20 Hz.

(2.471179929x10^20 Hz)/(2.410840467x10^42 Hz) = 1.025028393x10^-22

This is the same dimensionless ratio that has been previously found. This ratio is:

(R4/R2)^1/2 = (Tf)^1/2 = 1.025028393x10^-22

1.025028393x10^-22 (hc^5/3pi G)^1/2 = 2(electron mass)c^2 J

1.025028393x10^-22 (hc^5/3pi G)^1/2 = 1.637421013x10^-13 J

We can show that the maximum photon energy value (hc^5/3pi G)^1/2 J is reduced to h/(2pi)^2 J when subjected to the Tf time dilation factor.

Tf = (3hG/4pi c^5)^1/2 second / 2pi second

(hc^5/3pi G)^1/2 (Tf) = h/(2pi)^2 J

(hc^5/3pi G)^1/2 (3hG/4pi c^5)^1/2 (1/2pi) = h/(2pi)^2 J

This is an exact solution because the gravitational constant is canceled. Two energy values are precisely specified. These are labeled E2 and E3. The m value is electron mass.

E2 = 2mc^2 J

E3 = h/(2pi)^2 J

The E1 value is hypothesized to be the maximum photon energy. These energy values are related as shown. The E1 value is (hc^5/3pi G)^1/2 joule.

E1/E2 = E2/E3

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

4 m^2 c^4 = (hc^5/3pi G)^1/2 [h/(2pi)^2]

(G)^1/2 = (hc^5/3pi)^1/2 [h/(2pi)^2] (1/4 m^2 c^4)

G = 6.671745178x10^-11 N m^2/kg^2

This G value is shown to be consistent with the electron mass equations. This G value is closely related to the Planck constant.

In this concept, the gravitational collapse of a photon with wavelength (h/2mc) meters requires two phases. The first phase increases photon energy density. There is no change in photon energy because time dilation is matched to wavelength shortening. The photon wavelength (h/2mc) meter is shortened to the limit wavelength in this phase. The limit wavelength is (3pi hG/c^3) exponent 1/2 meter.

(3pi hG/c^3)^1/2 / (h/2mc) = 1.025028393x10^-22 m/m

While wavelength is reduced, time rate is reduced, so that available photon energy is constant. The next phase is the conversion of electromagnetic energy into gravitational field energy. In this phase two mass particles are materialized. The two particle reference frames spin at light velocity. The final resulting time rate at each particle radius (3Gm/c^2) will then approach zero seconds per second.

When an electron and a positron merge together, their reference frame angular momentum values are added. These opposite values will sum to zero. Each collapsed particle will have zero temperature only when spin is maximal. When angular momentum is canceled, the particles immediately become hot and radiate their energy away. The energy emitted in radiation is compensated for by a decrease in the mass of the particle pair.

Leonard Susskind has described a limitation on the smallness of things as a cutoff. A quote from Susskind follows.

"A cutoff sounds like a copout. Physicists have long speculated that the Planck length is the ultimate atom of space. Feynman diagrams, even those involving gravitons, make perfect sense as long as you cease adding structures smaller than the Planck length - or so the argument goes. This was the almost universal expectation about space-time- that it would have an indivisable, granular, voxelated structure at the Planck scale."

Evidence has now been presented showing that the limitation on the smallness of the electron is the radius 3Gm/c squared. When this limit is incorporated, the product of the Planck constant and the gravitational constant can be precisely defined.

h G = (6.62606957x10^-34) (6.671745178x10^-11)

A quote from Kenneth W. Ford follows.

"-- natural units h and c are arbitrary,-- but scientists can agree that they are directly related to fundamental features of the natural world. For an all-natural physics, we need one more natural unit, which has yet to emerge. This unit if it is found, may be a length or a time."

The natural length defined by this writer is the photon wavelength h/2mc meters, where m is electron mass.

h/2mc = 1.213155119x10^-12 m

When the derived G value is used, h/2mc is also equal to the value shown.

2pi (3pi hG/c)^1/4 = 1.213155119x10^-12 m

This wavelength uses the product of h and G to define a length value as proposed by Max Planck in 1900. A second quote from Kenneth W. Ford follows.

"Such a unit could usher in a whole new view of space and time in the subatomic world --- in layers of reality far below those explored so far."

The quotes are from the book, The Quantum World (page 28) by author Ford (2004).

Those readers who are interested in the quantized electron mass will want to read Chapter 8 from the book, Dreams Of A Final Theory, by Steven Weinberg (1992). Two significant statements are quoted below.

"--we think that all the forces of nature (including gravity) become united at something like the Planck energy,----." (page 206)

"But, because the standard model leaves out gravitation, we now think that it is only a low-energy approximation to a really fundamental unified theory and that it loses its validity at energies like the Planck energy." (page 207)

We now have evidence showing that electromagnetic and gravitational forces become united (equal) at the photon wavelength (3pi hG/c^3)^1/2 meter. A photon with this wavelength has the energy value (2/3)^1/2 times the Planck energy. This is the energy value labeled E1.

E1 = (2/3)^1/2 (hc^5/2pi G)^1/2 = (hc^5/3pi G)^1/2

The mass energy of the electron relates to two energy values. One energy value, labeled E4 is very small while the other energy value, labeled E5 is much larger.

E4 = (h/2)(1/2pi)^2 joule

E5 = (hc/12pi G)^1/2 (c squared) joule

(E4)^1/2 (E5)^1/2 = 8.187105066X10^-14 joule

(E4)^1/2 (E5)^1/2 (1/c squared) = 9.10938291x10^-31kg

The applicable value for the gravitational constant, G

is defined from the equation shown below.

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

G = 6.671745178x10^-11 N m^2/kg^2

This G value was previously determined from a number of separate evaluations. The dimensionless ratio, E4 divided by E5 has the value 1.050683206x10^-44 joule/joule. The same dimensionless ratio has the value seconds/second as shown below.

(3/2)^1/2 (Planck time)/(2pi seconds) = 1.050683206x10^-44 = seconds/second

I have described this ratio as a dimensionless time dilation limit. This ratio or the square root of this ratio shows up repeatedly in electron mass evaluations as shown below.

(1.050683206x10^-44)^1/2 (hc/12pi G)^1/2 = electron mass

The equation that defines a photon wavelength with energy specified by either the Planck constant or the gravitational constant is a significant part of this concept.

Energy = h f = h (c/wavelength)

Energy = (wavelength) (c^4/3pi G)

h (c/wavelength) = (wavelength) (c^4/3pi G)

(wavelength)^2 = (hc) (3pi G/c^4)

wavelength = (3pi hG/c^3)^1/2 meter

Energy = (2/3)^1/2 (Planck mass) c^2

Energy = (hc/ 3pi G)^1/2 c^2 joule

This upper energy limit provides a new interpretation of the Planck constant. Any photon wavelength will have energy that is equal to, or less than (2/3) exponent 1/2 times (Planck mass) times c squared. At any greater energy, the equation, E = h (frequency) does not apply.

Readers are asked to provide feedback to this writer if any error is found in equations presented here.