Ten Minutes for Physics
Number 11
Mass Growth with Motion and Matter Waves
We are exploring a theory of physics in which the universe is filled with a (rare) gas of Newtonian particles (which interact only by impacting each other).  A neutrino is a stable inhomogeneous state of the background particles.  Everything observed is either a neutrino or is produced by neutrinos.  
Each matter particle is made up of an orbiting neutrino whose tangential speed is the speed of light.  When a matter particle is at rest the orbiting neutrino making up the particle takes a circular path.  The electrostatic field produced by the neutrino is essentially spherically symmetric.
When a photon comes along and gets captured by the electrostatic field of the orbiting neutrino, the neutrino path changes from a circular shape to a spiral shape.  The neutrino speed still remains at the speed of light, albeit in a spiral path, but the matter particle that it makes up translates only as a result of its spiral path.  Figure1 shows the orbital path of a matter particle at rest and a particle moving at velocity "v".

(Insert Figure 1)
Figure 1.  Orbital Paths of a Neutrino Comprising a Matter Particle

Simply using the laws of conservation of mass, conservation of linear momentum, and the conservation of angular momentum several results are achieved.
When the photon interacts with the matter particle electrostatic field, part of the photon scatters and part of it is captured, as manifested by Compton scattering.  The part that is captured by the field adds mass to the particle and the mass varies with the velocity imparted by 	 where mv is the moving matter particle mass, mo is the mass of the matter particles when at rest, "v" is the velocity of the matter particle, and "c" is the speed of light.
This mechanism of matter particle motion shows a couple of significant points.  Since matter can only move by adding mass to a field which is rotating at the speed of light the impacted particle can never exceed the speed of light - no matter how much mass is added.  The equation also shows that as "v" becomes closer and closer to "c" that the mass grows without bound.	

Another significant point is when mass is added to a particle it is added off-center to the center of orbit.  Thus the mass added acts as an eccentric weight which causes the mass center of the particle impacted to undulate transverse to its path of motion.  The wavelength of this motion depends linearly upon the mass captured.  During impact, as a result of the conservation of angular momentum the smaller the mass is, it must be captured at a larger distance from the matter particle center of orbit.  Thus, the longer the wave length of the matter undulation.
The undulating motion as a result of mass capture phenomenon is observed for all matter.  Thus, electrons passing through slits display wave fringes just as photons do.
We will now derive the mass growth and matter wave relationships.
First we want to define "energy" and "kinetic energy" as used in the kinetic particle universe. We define energy of a particle simply as its mass times the square of its speed.  To define kinetic energy we apply a force "F" to a mass "m" and move the mass a distance ds (in the same direction as the force).  The work done on the particle is Fds (the force times the distance of movement).  The mass is accelerated and its mass times the particle acceleration is mdv/dt, which, of course, is equal to the force F.  Thus we can write 
Fds = (mdv/dt) ds
This expression can be written as 
	Fds = (mdv/dt) ds = mdv (ds/dt) = mvdv
We can integrate this to give the work
	
	

The term is the work done on the particle.  The term (1/2) is the final kinetic energy of the particle, and (1/2) is the initial kinetic energy.  The right hand side of this equation (1/2)(1/2) is the change in the kinetic energy of the particle.  Thus, the work done (which is energy) is equal to the change of kinetic energy of the particle.
	Energy can be transferred to a body in (at least) two different ways.  If a body of mass "M" is moving at velocity "v" and another (smaller) body of mass "m" also moving at velocity v (where the two velocities are parallel and in the same sense the energy of the combined body is Mv2 +mv2).  If a body of mass "Mo" is at rest and work is done (by an external force) to get its energy up to Mvc2 then the amount of work required is 2 times .  
	Let us now consider accelerating a matter particle by impacting it with photons where part of each photon is scattered and part is captured (by the orbiting neutrino making up the matter).  Let "p" be the matter particle linear momentum at any time t.  The linear momentum is the particle mass times the translational velocity "v" (of the particle mass center).  Due to a photon scattering we have the momentum change per unit time is	 and the change of energy (of the matter particle) is
 
In the limit
	.
The energy of the particle being impacted is mc2 since its mass is "m" (i. e., the mass of the neutrino) and its velocity, of course, is "c".  We can write the momentum in terms of "E" as
	
Solving for v from the above equation gives
	
and substituting into the above equation for E gives
	
or
	
Integrating this gives
	
where Eo/2 is the constant of integration.  Eliminating the factor 2 gives
	.
	When the particle is at rest, i. e., when its mass center is not moving the linear momentum is zero and E=Eo -- the rest energy moc2.
	The momentum can be eliminated from the above expression by using "cp" from , i. e., 
Thus 
		   .
  We now have
	
where
					( = v/c.
	Let Mv be the mass of the moving matter particle, which still consists only of mass moving at the speed of light, albeit in a spiral path (instead of the circular path of matter at rest).  Thus E=Mvc2 and Eo=Moc2.
thus
	Mv = 

This, of course is Einstein's famous equation for mass growth with velocity.  Note here, we have derived this result strictly from Newtonian mechanics.
	This equation clearly shows why matter can only move at speeds less than the speed of light.  However, this result is obvious from the construction of matter-which all consists only of neutrinos orbiting at the speed of light.
	This equation shows that when  all the mass (i. e., the orbiting neutrino) is moving at velocity c about the fixed center (of mass) and the particle energy is Moc2.  In order for the matter particle to accelerate, mass (or energy divided by c2) must be taken from each scattered photon and added to the matter particle field.  
	Let us now show why the addition of mass (at velocity c) to the field of a matter particle causes the matter particle mass center to take an undulating (i. e., sinusoidal path).  This concept is significant in that it relates a part of relativity (i. e., mass growth) to a part of quantum mechanics (i. e., matter waves). 
	Consider a proton impacted by a photon.  When mass "m" (photon energy/c2) is captured by the field of a matter particle the angular momentum added to the matter particle is  where h is , r is the half amplitude of the proton wave, v is the photon translational velocity, and m is the mass of the captured proton (i. e., energy divided by the square of the speed of light).
	When a photon is captured by a matter particle the photon angular momentum is added to the matter particle.  The angular momentum is  (twice the angular momentum of the neutrino).  The way their angular momentum is manifested is illustrated in Figure 2.  This figure shows the nominal

	(Insert Figure 2)
	Figure 2.  Path of a Moving Matter Particle Viewed from a Frame at Rest


(or average) straight path and the actual sinusoidal path of the center of mass of the particle-as viewed from a reference frame which is not moving.  
	If we view the matter particle from a frame moving at velocity v (i. e., so that the matter particle nominal translational velocity is zero) then the particle would appear as shown in Figure 3.

       (Insert Figure 3)
       Figure 3.  Matter Particle as Viewed from a Reference Frame Moving at Velocity v.

	The angular momentum of the photon mass which now is rotating about an axis fixed to the moving frame is , where r is its radius of rotation, m is its mass, and c is the velocity of light (of course).  
	For very low velocities we can write the momentum conservation equation for photon/matter particle interaction as 

 where Mo is the matter particle rest mass and v is its velocity.  Thus, we have
	
	We can use this relation to writeas
	
If we multiply both sides bywe get
	
Where h is the unreduced value of Planck's constant andis the distance around the path in the moving frame.  Again, for small m/Mo (i. e., for low energy interaction) the valueis the wave length.
	We now have derived the famous wave length equation
	
as postulated (but not derived) by de Broglie.
	This foregoing analysis relates the mass growth result from special relativity (mass growth results simply from adding photon mass to matter) to one of the fundamental postulates of quantum mechanics-(i. e., the de Broglie wave which matter has when it translates).
	It should again be emphasized that two of the foundational elements of modern physical theory (mass growth and matter waves) have been derived form classical mechanics.