Newtonian and Relativistic Equations involving Momentons
The momenton started out as a hypothetical particle, a fundamental "quantum of momentum".  (Why not?  Hasn’t everything else in Physics been quantized?)  Speculations involving the hypothesis deduced that it would contain a quantity of momentum that is the same absolute value in every reference frame (although different momentons may carry different amounts of momentum).  A momenton can also have a velocity, which would be different in different reference frames.  Note that because it is defined as being merely "momentum in motion", and is not energy in motion, it is not required to move at the speed of light.  There is an implication that the momenton can always be described with Newtonian simplicity.  This means that in an average reference frame its velocity might be less than that of light, but in some reference frames it may have a velocity equal to, or even greater than, that of light….  Next, the momenton would possess the kinetic energy of motion, this being the product of its quantized momentum and its reference-frame-dependent velocity.  Again, there is no requirement that a momenton move at light-speed, since it is not energy in motion; it is only momentum in motion.  Finally, as the hypothesis evolved, it became apparent that the force of Gravitation might be explainable in terms of quantum theory, as being the result of the exchange of virtual-momentons between inertial masses.  This means that momentons are actually gravitons, and we gain a fundamental explanation for the apparent equivalence of inertial and gravitational masses.  Also note that we are not really adding some brand-new hypothetical particle to the physicists’ zoo!

 

Definitions of symbols :m and m 0 for mass,
v for velocity of mass,
Ek for kinetic energy,
p for momentum of momenton,
s for speed of momenton (actually and also a velocity),
c for speed of light
 

Initial Newtonian Equations:  momentum of mass = mv, and its kinetic energy:  Ek = (mv2)/2
 

For a momenton, Ek = sp, always.
 
 

Initial Relativistic Equations:  momentum of mass =

 
 
 

and its kinetic energy:

 
 
 
 
 
 

The two questions to be answered are:

   (I)  If a momenton encounters and is absorbed by a stationary mass, what velocity will the mass acquire?

   (II)  What magnitude of a stationary mass can absorb a specified momenton?
 

Both the Newtonian and Relativistic solutions are derived below.
 
 
 

Since we are dealing with something defined as a "quantum" of momentum, ALL the momentum and kinetic energy of a momenton must be absorbed -- or none.
 

Newtonian answer for question (I):

If the stationary mass m acquires the all the momentum and kinetic energy of the momenton, then the mv of the mass will equal the p of the momenton, and the kinetic energy of the mass will equal the sp of the momenton.

If mv  and sp mv2/2,

then  smv  mv2/2,

mv  may be removed from both sides,

leaving  v/2,

which may be rearranged as  = 2s.

The mass will have a velocity equal to twice the original speed of the momenton.
 
 

Newtonian answer for question (II):

Using the results of the preceding answer,

if  mv  and = 2s,

then  m2s,

and simple rearrangement yields  p/2s.
 

One consequence of the all-or-nothing absorption idea is that nothing is likely to happen when a random momenton encounters an average mass.  With respect to virtual-momentons being used in a description of Gravitation, the rarity of absorptions automatically yields an extremely weak force....
 

The notion that a momenton must possess an exact combination of momentum and velocity, for it to be absorbed by a stationary mass, is subject to the Uncertainty Principle.  Uncertainty means that there will be some variation in any specified quantities of momentum and velocity (and consequent kinetic energy), at roughly the 14th decimal place of precision.  This degree of precision mandates derivation of the Relativistic answers to the questions, for use even at ordinary Newtonian velocities.
 
 

Relativistic answer for question (I):

As in the Newtonian answer, we begin by substituting the momentum of the momenton’s kinetic energy with the momentum of the mass, and setting the overall value equal to the kinetic energy of the mass:

Divide all terms by the mass, and multiply all terms by the “tau factor” (the square-root expression):

Divide all terms by the square of light-speed:

Rearrange:

Combine some terms on each side:

 
 
 
 

Square both sides:

Set a common denominator and then remove it:

Note that the first term on each side can be removed.  Then rearrange:

Combine the various instances of velocity of the mass:

Rearrange:

Simplify the fraction:

 
 

Final rearrangement:

 
 
 
 
 

For ordinary Newtonian velocities of a momenton, the expression (s/c)2 yields approximately zero, so the equation simplifies to the Newtonian answer.  If the momenton travels at light-speed, then the absorbing mass would also travel at light-speed (caveat:  note the magnitude of the absorbing mass, when the answer to the second question is derived).  Finally, if the momenton travels faster than light, then the absorbing mass will nevertheless move at less than light-speed.  Intriguing....
 
 

Relativistic answer for question (II):

Again we incorporate the answer to question (I), as the momenton’s momentum is set equal to the relativistic momentum of the mass:

 

Rearrange; begin preparations for removing layers of divisors:

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Remove one division layer; prepare for more:

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Remove more division layers:

 
 
 
 
 
 
 
 
 

Move some terms into the square-root operator; cancel that tail-end light-speed-squared:

 
 
 
 
 
 

Deal with the ‘1’:

 
 
 
 
 

Simplify and expand:

 
 
 
 

Simplify:

Simplify, and take the square root:

 
 
 
 

Final simplification:

 
 
 
 

As before, at ordinary Newtonian velocities the expression (s/c)2 yields approximately zero, after which the overall equation simplifies to the Newtonian answer.  For light-speed momentons, only zero-mass objects can absorb them, so photons in a gravitational field can keep moving at light-speed, but change direction because they acquire momentum!  And although the value of light-speed is different in a gravitational field than it is in empty Space -- the net effect is the same!  Meanwhile, and most intriguing, is the observation (from this equation) that for momentons which move faster than the speed of light, only negative masses can absorb them!  This is neat for two reasons: (A) It is consistent with other places in Physics where faster-than-light has been associated with negative mass/energy; (B) If negative mass/energy can exist, virtual negative mass/energy will exist.  Then the enormous "vacuum self-energy" of all virtual particles suddenly becomes Zero, and a tough problem in making Quantum Mechanics compatible with General Relativity is solved (the vacuum self-energy will have no associated gravitation).
 

Questions?  Comments?  Math errors?  Let me know!  Vernon Nemitz  Note that the fundamental speculations leading to these equations are described and linked here.
 

P.S.  Note that if gravitons actually are the momentons described here, then consider the Time Reversal Symmetry of Quantum Mechanics.

Simply because a sitting mass might absorb a momenton, it ought to be possible to make a mass emit one.  Note also that the very
definition of a momenton implies that it can have a high ratio of momentum to (kinetic) energy.  So compare a beam of momentons to a
beam of photons, and think about Action and Reaction.  The momentum carried by that beam must be balanced by overall motion of the
beam-generator, and the effect might be very significant!  Thus, a method of "propellantless propulsion" would exist.  That would bode
well for the future of space development.  My brother, Gregory Nemitz, is a strong supporter of space development, and he is the Webmaster
of  http://www.nemitz.net.   As part of his efforts to promote space development work (besides putting up with my wild ideas), he also
sells Final Frontier Beef Jerky.