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 p = mv and sp = mv2/2,
then smv = mv2/2,
mv may be removed from both sides,
leaving s = v/2,
which may be rearranged as v = 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 p = mv and v = 2s,
then p = m2s,
and simple rearrangement yields m
= 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:
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Divide all terms by the mass, and multiply all terms by the “tau factor” (the square-root expression):
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Divide all terms by the square of light-speed:
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Rearrange:
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Combine some terms on each side:
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Square both sides:
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Set a common denominator and then remove it:
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Note that the first term on each side can be removed. Then rearrange:
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Combine the various instances of velocity of the mass:
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Rearrange:
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Simplify the fraction:
 |
Final rearrangement:
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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:
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Remove one division layer; prepare for more:
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Remove more division layers:
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Move some terms into the square-root operator; cancel that tail-end light-speed-squared:
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Deal with the ‘1’:
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Simplify and expand:
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Simplify:
 |
Simplify, and take the square root:
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Final simplification:
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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.