An approach to particles from relativistic kenetic energy (incomplete draft)

5. Relativity/quantum bridge, the gravitational coupling constant and the quantization of velocity

From Section 4, let

G(s) = G((1- v^2/c^2)^-1/2 -1) + (v^2/c^2)(1- v^2/c^2)^-3/2)

(quantum level only) so that the vis viva equation becomes

v^2 = G(s)(m1 + m2) / r (12)

where in a planetary model v is the tangential velocity of a quark or nucleon m2 relative to quarks or nucleon(s) m1, and r is the distance between mass centers.

As with electrons in atoms let

n L(s)/2 = 2 pi r (13)

where L(s) is spacetime wavelength, and accounting for the fundamental wavelength,

n = 1,2,3 ...

It can be shown that

E(s) /P(s) = c^2/v. (14)

Combining (7), (9) and (14)

L(s) = h / P(s) (15)

where L(s) suggests a deBroglie wavelength. Combining (13) and (15)

p(s)r = n/2 (h/2 pi) (16)

expressing angular momentum. Eliminating r in (12) and (16),

G(s)P(s) / v^2 = (n/2) h/2 pi / (m1 + m2)

Let V1 = (((1 - v^2/c^2)^-1/2 -1) + (v^2/c^2)(1 - v^2/c^2)^-3/2)

V2 = ((1 - v^2/c^2)^-1/2 -1)

so that V1*V2/v = (n/2) ((h/2 pi) c) / G(m1 + m2)

which reduces to

(1 - (v(n))^2/c^2)^-1 = (n/2) ((h/2 pi) c) /G(m1 +m2)m2 (17)

= (n/2) Ag^-1 (17a)

where the left side has a relativistic character, and the right a quantum -- the inverse of the gravitational coupling constant (Ag) when the mass term is a proton squared. Plots of the complete and reduced velocity functions remain close to zero until v -->c, where they tend to overlap. Equation (17) can be termed the "relativity/quantum bridge," the possible importance of which cannot be over estimated.

6. Isolated particles, and the strong force

In Equation (16) when n = 1,3,5 ..., there is an indication of fermions, while n = 2,4 ..., bosons.

For an isolated spinning particle the force maintaining particle integrity is either

Fg = ((G (m/2) (m/2)) / (h/mc)^2 )

(((1 - v^2/c^2)^-1/2 -1) + (v^2/c^2)(1 - v^2/c^2)^-3/2)

or Fg = ((G (m/2) (m/2)) / (h/mc)^2)

(1.3 (1 - v^2/c^2)^-3/2) v -->c (lemma)

from Section 4 where the velocity term in the reduced version overlaps that in the original expression in a plot at v -->c. The distance term is from (8). Combining the above and (17),

Fg,n = ((n/2)^1/2 Ag^-1/2 + (n/2)^3/2 Ag^-3/2)

((Gc^2)/4h^2) m^4 (18)

or Fg,n = (n/2)^3/2 (0.058) (c^7/Gh)^1/2 m. (18a)

If m is that of Equation (11),

Fg,1 = ((1/2)^1/2 Ag^-1/2 + (1/2)^3/2 Ag^-3/2)

(1/4) ((G^2 Hc^4) / h^4)^1/3 (18b)

or Fg,1 = 0.021(c^17 h H^2 / G^5)^1/6 (prediction) (18c)

the force perhaps providing quark integrity, and the Strong force.

7. The unit charge

In a given charged particle let

Fg,1 approx > Fcoulomb (lemma)

In Equation (18) when m is the Planck mass and n = 1, and r1 is h/mc from Equation (8), the charge, q, of the above is about the unit charge,

10^-18 > q > e, (proof)

also substantiating Section 6.

8. The zero rest mass particle, and the Weak force

There appears to be a complementarity between spacetime waves and photons; from Section 4,

E(s) = hV(s) v --> c (8)

and commonly E = hV, though not yet established in the present context (being axiomatic, see below).

Thus

E(s) / V(s) = E / V = h. (lemma) (19)

Let E(s') = -i h V (v^2/c^2 -1)^-1/2 (preliminary)

be a photon spacetime wave, where v is velocity of expanding spacetime about the entity. However, spacetime waves for non-zero rest mass particles were developed from pure relativistic kenetic energy, where it was not assumed the particle had any independent existence beforehand; therefore

E(s') = -hV -i hV(v^2/c^2 -1)^-1/2 (lemma) (20)

would seem more proper. Implicit in this relation is the energy of expanding spacetime about the entity that ultimately would exist only at v = c. Velocity , v, in this relation is that of spacetime itself about the photon precursor; the energy of the photon precursor, then, is non-local, as with non-zero rest mass particles.

It can be shown that

E(s) / V(s) = E(s') / V(s') = h V L / v

(where V(s') = v / L(s') = v / iL(v^2/c^2 -1)^1/2)

for v --> c as about a primal event, therefore

L V = c (proof)

where V is the frequency of a common photon (establishing (19), and confirming that a photon spacetime wave of (20) has a velocity of c (relative to expanding spacetime).

When the v < c form of (20) is employed in conjunction with (17) this might be seen as a W particle, agent of the Weak force.

9. Conclusion

Thus the fundamental elementary particles and four forces are traceable to gravitation, and gravitation depends on accelerated universal expansion -- an inertial effect. This could explain the experimental equivalence of inertial and gravitational mass; apparently there is not only an equivalence but also an identity.

(end of An approach to particles from gravitation)