© 2003 By Michael Harney

It has been shown in "Derivation of Hubble’s Constant and the Quantization of the Gravitational Field" and from Mach’s principle that the rest-energy of any object is equivalent to the universal gravitational potential energy acting on that object [1]. It can also be shown that one simplified quantization scheme for the gravitational field starts by assuming the photon is a standing matter-wave of fundamental wavelength in a two-dimensionsal box (more on this simplification later) with dimensions equal to twice the radius of the universe. In this process, we solve Schrodinger’s equation for this problem and find a solution for all allowable energies that the particle can have as follows [2]:

2p ²h²/(2m_pL²) = m_pc² ,(2)

m_p = [2p ²h²/(2(2r)²c²)]^1/2 = 1.8 x 10^-68 kg ,(3)

This value for photon mass closely matches an estimate proposed by JP Vigier in an analyis of what was considered experimental error in the Michelson-Morley experiment [5]. The Michelson-Morley experiment actually yielded a small ether drift of 8 Km/sec that was much smaller than the expected 320 Km/sec and the small drift was attributed to experimental error. An analysis of the data by Morley and others however, has shown that the "error" is periodic with respect to the rotation of the Earth and it’s periodicity is verified in other multiple experiments. Vigier has proposed a photon mass of 10^-68 Kg from the Einstein-DeBroglie relation that will offset the ether drift of 8 Km/sec and restore a relativistic outcome for the experiment (otherwise, the ether drift indicates an absolute reference frame).

The underlying principle that is used for the above derivation of (2) and (3) is that rest-mass energy is equivalent to quantum well energy and we have shown from [1] that Universal gravitational potential energy (UGPE) is equivalent to rest-mass energy so quantum-well energy is then equivalent to UGPE, which allows us to quantize the gravitational field as follows:

[(n_x)² + (n_y)² ] p ²h²/(2mL²) = m M_uG / r ,(4)

Where r = radius of universe, M_u = mass of universe (1.44 x 10⁵³ Kg), and m = mass of object

For any particle with mass m, there is a formula for the quantization of mass due to equations 1 and 2 as follows,

m = [((n_x)² + (n_y)² ) p ²h²/(2(2r)²c²)]^1/2 ,(5)

Where, for the photon the quantum numbers are 1 as we have just shown and for all other particles the quantum numbers are greater than 1. For the highest known mass, the mass of the universe, we solve for the combination of ((n_x)² + (n_y)²) by taking a ratio of two equations modeled after (4). By leaving the quantum numbers as variables in the numerator for the mass of the universe and setting them equal to 1 in the denominator for the mass of the photon (with all but quantum numbers cancelling on the left hand side),

((n_x)² + (n_y)²) = M_uc²/ m_pc² ,(6)

Where M_u = Mass of universe and m_p = mass of photon

Or ((n_x)² + (n_y)²) = 10¹²²

And the effective quantum number (n_eff, the quantum number of the same order of manitude as either n_x or n_y) is,

n_eff = [((n_x)² + (n_y)²)]^1/2 = 10⁶¹ ,(7)

The physical interpretation of these results is that rest-energy (and hence rest-mass, because c cancels from numerator and denominator in (6)) is quantized and can assume only certain values. Likewise, a particle of higher mass than the photon will have a higher n_eff. Also, n_eff is the wave number from the solution to Schrodinger’s equation for the probability density function. For n_eff = 1, the probability density function assumes a half-sine function with a peak amplitude at r = 0, or the center of the universe [3].

Likewise, a particle of higher mass than the photon will have a higher n_eff, which corresponds to a shorter DeBroglie wavelength, which for the standing waves that we have assumed in (1) are given by

where L = 2(radius of universe). This last development corresponds directly with Heisenberg’s Uncertainty principle,

(D E)D t = h ,(9)

Where D E is found from (1) as,

and D t = D l /c ,(11)

where c = speed of light and D l is found by differentiating (8) with respect to n:

D l = -4L/(D n_eff)² ,(12)

Therefore, combining (10), (11) and (12) and recognizing L = 2r,

(D E)D t = [(D n_eff)² p ²h²/(2mL²)][ -4L/c(D n_eff)²] ,(13)

= h²/(mrc)

This shows that an increase in energy (by increasing mass as in (10)) causes a decrease in D t (12), which makes (D E)D t constant. Therefore, Heisenberg’s uncertainty principle is derived from assuming a standing wave formula for all masses (with n = 1 corresponding to l = r) and applying Schrodinger’s equation to calculate the energies in the standing waves. A photon has the lowest location resolution and the highest momentum resolution (it’s mass and velocity being known very precisely). The low-resolution of photon location (let’s call it non-locality) may explain photon entanglement, where photon’s initially linked by metastable quantum states are capable of non-local communication.

As an object approaches the speed of light and it’s relativistic energy approaches M_uc², it’s n_eff approaches 10⁶¹ and it’s matter-wavelength is reduced. This may be where the Planck length is most commonly noticed – as a limiting factor on n_eff which in turn limits the velocity that an object can attain as it approaches the speed of light.

Now we examine how quantization affects the vector particles and their ranges for the four forces. The Planck length is defined as,

L_p = [(hG)/(2p c³)]^1/2 ,(15)

Therefore,

Where the constants have been previously evaluated [4].

(r_u)/ (L_p) = n_eff = 10⁶¹ ,(16)

Therefore, the maximum quantum number (n_eff = 10⁶¹) that corresponds to the ratio of the maximum to minimum rest-masses in the universe is also equal to the ratio of maximum to minimum radii that the gravitational force acts over. By incorporating (6) and (16) together we find,

GM_u/r_u² = Gm_p/L_p² ,(17)

Or that the gravitational force due the Mass of the universe acting on an object over the radius of the universe is equal to a photon’s gravitational pull on the same object acting over the distance of the Planck length. This simply states that the electromagnetic force and its vector particle the photon has the same mass/range relationship on its minimum scale (Planck Length) as the gravitational force has over its maximum scale (mass of universe, radius of universe). It is also interesting to note that the relationship in (17) applies to the strong nuclear force as well:

GM_u/r_u² = Gm_pi-meson/ r_strong ² ,(18)

Where m_pi-meson = 139.6 MeV and r_strong is the distance the strong-nuclear force acts over (or the maximum nuclear radius), which after solving (18) we find

r_strong = 7.88 x 10^-15 meters,

which is the typically the maximum nuclear radii and the range over which the pi-meson acts in strong interactions. For the weak nuclear force, which has a generally accepted range of 10^-18 meters and using the constant evaluated in (17), the estimated mass of the vector particle (which is mostly likely the electron-neutrino) is 4 x 10^-36 Kg or 2.2 eV/c².

Based on (2) where we assume that the rest-mass energy of a particle is equivalent to it’s quantum-well energy we formulate the conclusion that a particle can be viewed as a series of standing-wave ripples in the fabric of space-time, where the amplitude of the ripple is mc² and this amplitude is also the vector axis of particle motion. The cartesian-coordinates that are normal to the particle’s axis of motion correspond to the DeBroglie wave vectors specified by n_x, n_y and L in (1). Also, as we have calculated a mass for the photon in (3) and assigned n=1 to it’s DeBroglie wave while it travels at free-space velocity, we know that it’s E and H vectors are normal to it’s direction of propogation and therefore the E and H fields are mapped into the same plane as it’s DeBroglie waves. The amplitude of the photon’s DeBroglie waves are found to be m_pc² which is also equivalent to it’s UGPE as defined by (4), which allows for future derivations between the gravitational field and the electromagnetic fields.

If the particle is thought of as a ripple in this space-time fabric, then classical mechanics should apply to this fabric. If we take a simple approach of using Hooke’s law for an elastic medium, then the energy stored in a compression or tension of space-time fabric is:

Where D r is the change in distance due to compression or tension (which in turn is the distance between objects in the fabric of space-time) and k is the elasticity constant. But this compression or tension translates into changes in rest-mass energy,

So combining (19) and (20),

The term m/D r² has been found from (17) and (18) above to be a constant for the range of particles and forces in their limited or extreme case and when combined with (21) yield,

k = 7.18 x 10¹⁷ Newtons/meter ,(22)

As (21) is based on the integration of Hooke’s force law, it should also be true that F = kx, which is the force of compression or tension in the spacetime fabric and we find that this is the case for the boundary condition of x = radius of universe as follows:

F = kR_u = G(M_uM_u)/[(R_u)²] ,(23)

Where M_u is mass of universe, R_u is radius of universe and k = 7.18 x10¹⁷ as found above. Equation (23) simply states that the force due to the mass of the universe (M_u) acting on itself over its own radius is equal to k multiplied by a space-time displacement. This space-time displacement is then equal to the radius of the universe. This is the overall stretching force of the universe corresponding to its mass and it produces a pulling force that can produce standing-wave ripples (whose amplitude is the rest energy, mc² of a particle) in the fabric of spacetime. For any material of mass/unit-length u, where we introduce a force along one axis in the material we get waves of speed v as follows:

v = (F/u)^1/2 ,(24)

We know from (23) that F = kR_u for the universal tension and we know the mass of the mass per unit length of the universe as M_u/R_u so (24) becomes

v = (kR_u² / M_u)^1/2 ,(25)

For R_u = radius of universe, M_u as mass of universe and k as found above it is shown that v = c, showing that the speed of a wave in the space-time fabric produced by the universal tensile force in (23) is equal to the speed of light. Note that v is not the speed of the mass itself but rather the speed of its Debroglie wave which we assume travels in the plane normal to the particle’s velocity vector. We assume this because the rest-energy of the particle can be set equal to the quantized energy in a two-dimensional quantum well and then the relation in (21) is derived. This then defines the rest-energy of the particle from its mass as mc² which is based on the energy in its Debroglie wave, which is proportional to v² when v = c. The particle may be moving at some speed and this produces separate, non-standing DeBroglie waves based on the particle’s speed but in (24) we propose that standing DeBroglie waves normal to the velocity vector is what defines the rest energy of the particle. This is consistent with the definition of mass as an inertial quantity that is resistant to force, where all force vectors can be replaced with one resultant vector that is valid in only one axis. Hence, our model for mass (which we define as the amplitude of standing DeBroglie waves) is valid in only one dimension (while the other two dimensions define the plane of DeBroglie wave propagation).

From (25) we know that the upper limit for v is c and this corresponds to a tensile force equivalent to the mass of the universe, therefore smaller masses will exhibit a force that is smaller than (23). If we take a given mass of say, 10⁶ Kg and calculate from (21) what the displacement x is (knowing k from above) we find x = 500.6 meters. This is the average stretch in spacetime that corresponds to this mass and because the force is smaller than in (23) we expect the velocity of the DeBroglie waves to be smaller than c which will amount to time-dilation effects at distance x from the gravitational source. When we substitute x = 500.6 meters into (25) in place of R_u we find v = 0.73 cm/sec, which is the velocity component of the DeBroglie waves moving towards the mass in the quantum foam at 500.6 meters from the center of the mass. There is also the pull on the DeBroglie waves from the remaining mass in the universe (kR_u) which is similiar to Mach's principle, and results in a velocity of c in a radial direction away from the mass. The net velocity is obiously c - 0.73 cm/sec which is very close to c.

The time dilation effects result from the displacement of the deBroglie waves as they travel relative to c. When we take the ratio of the inward radial velocity (toward the mass source) to the velocity c that results from universal pull (kR_u) for the given example we find v/c = 2.43 x 10^-12 and

(v/c)² = 5.90 x 10^-24 ,(26)

In general relativity the time dilation at a distance R from the center of the gravitational source M is

T = T₀ / (1-2GM/(Rc²))^1/2 ,(27)

For the mass of 10⁶ Kg and R = x = 500.6 meters , 2GM/(Rc²) = 2.96 x 10^-24 = (1/2)(v/c)² where (v/c)² is found from (26). Therefore, the time dilation formula of (27) when combined with the DeBroglie velocity waves of (26) and (25) becomes,

T = T₀ / (1-.5(v/c)²)^1/2 ,(28)

which excluding the factor of 0.5 which may be an averaging effect, is the standard Lorentz transformation for time dilation of a mass moving at velocity v. So we find the time dilation effects due to general relativity (27) are essentially the same as those of special relativity (28), when the velocity of DeBroglie waves are considered. Although we used a specific mass for this example of m = 10⁶ Kg, it can be shown in general by combining (25) and (21), and substituting x = R that

0.5(v/c)² = 2GM/(Rc²) ,(29)

We know from (25) that the velocity of the DeBroglie waves under universal tension is a function of R_u:

v = c = R_u (k / M_u)^1/2 ,(30)

We also know that R_u is increasing as a function of time so v depending upon the sign taken for (k / M_u)^1/2 , we know v must be increasing or decreasing as a function of time. From previous analysis by this author[1] it was shown that the speed of light is decreasing as function of time by a similar relationship as in (30). Therefore, we take the sign as negative and the derivative of (30) shows:

dc/dt = -[(k / M_u)^1/2] (dR_u/dt) ,(31)

and because dR_u/dt = c,

dc/dt = -[(k / M_u)^1/2]c ,(32)

We can write the solution to this differential equation of the first order as:

c(t) = c(0)e^(-Ht) ,(33)

where

H = [(k / M_u)^1/2] = Hubble’s constant of 2.23 x 10¹⁸ sec ^-1 which is consistent with the results found in the previous analysis of c-decay[1].

The quantization of mass and gravitational fields can be interpreted by viewing these particles or fields as standing DeBroglie waves in a space-time fabric. The potential energy in this fabric is equivalent to the rest-energy of particles and the potential energy of gravitational fields. The standing DeBroglie waves of a particle are projected in two-dimensions normal to the axis of motion of the particle. The E and H vectors of electromagnetic fields are mapped into the same plane as the DeBroglie waves, with interaction occuring as energy displacement inside the space-time fabric. By applying Schrodinger’s equation to this standing wave with the two-dimensional plane that it is in, the mass of particles can be interpreted from the quantized energy of the waves.

[5] Vigier, J.P. "Relativistic Interpretation (with Non-Zero Photon Mass) of the Small Ether Drift Velocity Detected by Michelson, Morley, and Miller", APEIRON Vol.4 Nr.2-3, pp.71-76, Apr.-Jul 1997.