[[Mod. note -- I apologise for the long delay in posting this article,   
   which was received by by s.p.r moderation system on 2019-01-28.   
   -- jt]]   
      
   On Friday, August 24, 2018 at 10:40:51 AM UTC-4, toadast...@gmail.com wrote:   
   [[Mod. note -- 88 excessively-quoted lines snipped here.  -- jt]]   
   > 24-AUG-2018   
   >   
   >   
   > Gamma-limit Conjecture Briefly:   
   >   
   > The coupling, between the energy-momentum of a mass m, and constant   
   > scalar curvature S, of relevant 3-sphere, may become effective if   
   > a) the mass can be confined to within twice its Compton wavelength,   
   > and b) the particle's velocity satisfies the condition gamma = 1/2(S),   
   > where gamma = (((1 - v/c)^2)^1/2)^-1.   
   >   
   > According to Friedrich, a solution of the Einstein-Dirac equation is   
   > a field psi, satisfying the equations,   
   >   
   > Ric-(1/2)S*g = T_psi ; D(psi) = lambda(psi),   
   >   
   > where Ric is Ricci tensor, S = Tr(Ric) is scalar curvature, g is   
   > metric tensor, T_psi is energy-momentum tensor, D is Dirac operator   
   > and lambda is WK-number.  Specifically, for the two metrics with WK-   
   > spinors on S^3, contained in the space N^3(K,L,M), where K = M = 1   
   > and L = 1/4S, I find   
   >   
   > psi = S / lambda ; lambda = S*psi^-1; S = lambda*psi = D(psi).   
   >   
   > Now, with gamma-limit defined as (gamma - 1) = (1/gamma), we put an   
   > electron with mass m, in a square potential with dimension   
   > d = 2(h/mc) and make the electron velocity 78.6151385% light speed.   
   > The electron's kinetic energy is E_n = (1/2)mv^2 and we then find the   
   > total energy is E = E_nS. . .  At the gamma limit, the electron's   
   > energy is the product of the kinetic energy and the scalar curvature.   
   >   
   > In the inelastic scattering model, an incident photon gives the electron   
   > in the box its kick, the incident photon's frequency is   
   >   
   > nu_i = (E/h)*(1/2), with h Planck's constant (eV sec).   
   >   
   > If the diameter of the potential is made about 38% smaller, the incident   
   > photon frequency has to be corrected by another factor of S/2,   
   >   
   > nu_i = (E/h)*L; L = 1/4S.   
   >   
   > Just beyond this scale and gamma diverges.  To within a factor of 4   
   > the input energy is E_i = 2(E).  So maybe pair production to account   
   > for the gamma divergence.  Dunno.   
   >   
   >   
   > Cheers,   
   > mark jonathan horn   
      
   28-JAN-2019   
      
      
   I noticed, in a recent review of notes, that the mass term I defined was   
   NOT that of the electron, but rather the extrapolated   
   photocharge mass from an inelastic scattering model, found   
   to account for experimental data at optical energies from a   
   camera's output.  Pretty sure any mass works at the gamma-limit,   
   but haven't done the calculations.   
      
   As a point of clarification then, the photocharge mass is:   
      
   m = (m* - m_e),   
      
   where effective mass m* = m_e(1 + eta^2)^1/2;   
      
   m_e is electron mass and eta is electric field strength parameter[1]   
      
   eta = e( / (m_ewc)) = (gamma)(beta).   
      
   Here e is electron charge,  is r.m.s. electric field   
   strength and w is angular frequency.   
      
   I haven't re-computed for any other mass, and lazily   
   suspect the gamma-limit is generally sound for any   
   mass.  I'm presently lacking the fortitude to spelunk   
   into this conjecture to see why e.g. there would not   
   be an infinite spectrum of stable masses.   
      
   Maybe someone with a torch can say how deep this is.   
      
   That said, the salient point appears to be, at the gamma-limit:   
      
   m = E_n , c = (S)^1/2,   
      
   with E_n = 1/2(m_ev^2).   
      
   Mass as kinetic energy confined, liberated at the gamma-limit   
   with reduction of radius of confinement (like, in a collision).   
      
   Speed of light equal to square root of the constant scalar curvature.   
   I dunno.  This just makes for an intolerable neural itch.   
      
   Cheers,   
   mj horn   
      
   [1] Fernow, R.C. et al 1989 "Proposal for an Experimental   
   Study of Nonlinear Compton Scattering"   
   www.bnl.gov/atf/experiments/References/altprop.pdf   
      
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