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Two Conclusions of Standard Physics Reconsidered

Infinities in Mathematics are an opening of the imagination. Unfortunately, infinities in Physics are a hint that a physical phenomena is being modeled incorrectly. In this section I will explore two infinities which I consider “elephants” in physics. They are in the room and are obvious but are psychologically overlooked.

1. The Uncertainty Elephant: (see section 3 and section 6 and section 12)

Digital Wave Theory considers Heisenberg’s uncertainty principle as a misreading of how motion occurs on the quantum level. The elephant in the room is not the uncertainty principle itself, which in a vague way makes sense and will be hard to get rid of because it provides a convenient “rug” that a lot of nonsensical notions in physics can be swept under.

The elephant created by the uncertainty principle is the infinity of uncertainty created when a particle has no motion. The uncertainty is formulated as ΔxΔp ≥ h/2 and when a particle is stationary the Δp is 0 and the uncertainty in position Δx goes to infinity. So an electron that is not moving can be anywhere in the universe!  DWT thinks this is goofy. Instead DWT says:

That once it is recognized that an electron moves by hopping (appearing and disappearing) along its wavelength and is never stationary, its motion is predictable and not a fundamental unknowable. This motion of quantum mechanical particles can be verified experimentally by following a Buckyball C60 as outlined in the experiment in Section 14.

2.  The Special Relativity Elephant:

In special relativity, relativistic mass is defined as: mr = m0 /sqrt(1 - v2/c2). With this relationship, as the velocity of a particle (an electron) approaches c its mass approaches infinity. This is goofy and I believe that Einstein also knew it was nuts.

In contrast to this, DWT says that an electron (or any quantum mechanical particle) is limited as to how close to the speed c it can get.  Once the mass of the particle gets to the Planck Mass (2.176 51(13) × 10−8 kg) it can go no faster. This conclusion can be understood via Section 31 .

The paper entered in the 2013 FQXi essay contest “An Elephant in the Room” goes into the details about how mass gets cut off from going to infinity. It is reproduced below:

An Elephant in the Room

Abstract: There is an elephant in the room.1 Something obviously is wrong and no one pays any attention to it. We have been taught to believe that the mass of an electron (or any particle or even a golf ball) can approach infinite mass if its velocity approaches the speed of light. This phenomena (aka the elephant) is represented by the formula:  M = m0/(1-v2/c2 )0.5.

      This mistaken truth is in a lot of textbooks even though it can never be proven because it is impossible to prove by experiment that anything goes to infinity. Nowhere do we see massive quantum mechanical particles larger than 22 micrograms whatever their velocity, yet physicists as a group believe in them. This strikes me as just plain goofy. A case will be made that any particle can only get to a maximum mass of 22 micrograms (a Planck mass). This is about the mass of an eyelash and is a lot less than infinity.

     Since ordinary classical masses (i.e. golf balls) are composed of particles they too will not approach infinite mass as they approach the speed of light. It may seem that getting rid of “the elephant” would be a major alteration to the theory of special relativity. However, the theory remains intact (but without the elephant). At the end of this essay a way of looking at why there is a limit on particle mass will be suggested.

1. Preamble: There is controversy over the equation M = m0/(1-v2/c2 )0.5. Einstein and others preferred to use the viewpoint of momentum instead of mass2 where the momentum P is equal to     Mv = m0v/(1-v2/c2 )0.5. The P represents momentum. I like to think of the P as indicating “Pachyderm”. Einstein knew the elephant was “ugly” and tried to dress it up a bit by noting that the thing that made the elephant ugly was the distortion of “space-time” caused by the velocity of the object. The intent was to look at the phenomena as space-time changing and not mass increasing. This distinction is a good one if one bothered to check out if space-time keeps on changing when v gets very close to c, but no one has bothered. For the purposes of this essay unlimited mass increase, and unlimited momentum increase, and unlimited space-time compression (or curvature) are the same elephant because they directly or indirectly violate the physical limits on smallness.

I anticipate many objections to this essay primarily because It questions many concepts that are currently taught in physics classrooms and are held to be sacred. These objections will be covered in the final section on “Objections”. This work does impact one of the results of special relativity (the elephant) but the core concepts of special relativity are intact. Einstein was the rare genius who not only made outstanding contributions to physics but also helped others make them. In this essay I use the special theory of relativity and hold the equation M = m0/(1-v2/c2 )0.5 as completely valid.

Wait a minute! Didn’t you just say that M = m0/(1-v2/c2 )0.5 is the elephant and now you are saying it is completely valid. What gives?

As an equation it is completely valid. However, it has an unexpressed assumption that the velocity “v” can approach “c” to any precision we chose. This assumption will be shown to be untrue. So,        M = m0/(1-v2/c2 )0.5 is completely valid and has limits that have not been investigated. Let’s look more closely at how close to the velocity of light v an get.

2. The Basic Building Blocks:

1. The Compton wavelength is a valid formulation for the mass of all quantum           

    mechanical particles.3    

2. Particles (aka sub atomic particles) are what make up ordinary matter. Their salient

    characteristic is that they have the wavelike property of interference. The most    

    massive particle to date that has been tested for the interference property is the          
           Buckyball C60.4

3. The Planck length is the shortest possible Compton wavelength.5

3. Planck Units:

The usefulness of Planck units is their fundamental nature.3 They are composed of the constants of nature G, h, c which are considered changeless, and therefore the Planck units themselves being composed of these constants of nature are also changeless. The standards organization NIST6 is very confident in this changelessness and is attempting to get all measurements traceable to constants of nature.7,8

In the analysis to follow I will be using the Planck length PL and the Planck mass Pm as  fundamental limits that are unchanging irrespective of motion. They are immutable constants that are the same in all frames of reference. This means they are unaffected by relativity and that no matter the relative velocity, PL and Pm remain constant. Note that the Planck Length is PL = (hG/c3)0.5  and the Planck Mass is Pm = (hc/G)0.5 .9 See the glossary at the end of this essay for more information.

4. A Fundamental Limit on Particle Behavior:

The shortest wavelength a particle can have is the Planck length.5 There is no proof or derivation for this. It is an axiom that seems to make sense in that the Planck length is a very small number (1.616 ×10−35) that is defined only in terms of constants of nature. A small special number not effected by relativity what could that mean? The limit of measurable length, sure why not.

5. The Compton Wavelength:

Definition: The Compton wavelength (λc) of a particle is equivalent to the wavelength of a photon whose energy is the same as the rest-mass energy m0 of the particle.10

A Visualization: When a photon is trapped between two mirrors it takes on the appearance of a particle. If the spacing of the mirrors is the Compton wavelength (λc), then the photon moving back and forth between the mirrors (at the speed c) has an energy that is equivalent to the rest mass m0 of the particle. This is just an aid in understanding. I have no idea why space-time should form a set of mirrors spontaneously at particular wavelengths. However it makes a useful visualization. See the diagram in the next section.

Deriving the Compton Equation: 11

a. The Einstein-Planck relation is E = hf and is the source of the Compton wavelength.

b. This can be written E = h/(cλc)

c. The energy of the photon trapped at the Compton wavelength is m0c2.

d. Setting the two energies equal we get: m0c2 = h/(cλc)

d. Solving for λc we get: λc = h/(m0c) The equation for the Compton wavelength.

6. The Compton Wavelength as a function of Velocity.

a. The Compton wavelength of a particle is given by: λc = h/(m0c)

b. λc is a distance and is subject to relativistic compression.

c. Using the Lorentz transform we get that λc = [h/(m0c)](1-v2/c2 )0.5

       d. This equation can be rearranged as:  λc = [h/(cm0/(1-v2/c2 )0.5)]

e. We notice that the term m0/(1-v2/c2 )0.5 normally associated with the elephant is in the            
            denominator. When v = c and the mass goes to infinity the Compton wavelength λc

       goes to 0. However, λc is not allowed to get to 0, because the shortest value for λc is

     the Planck length. This limitation on λc is also a limitation on the velocity v.

f.  The velocity of a particle will be made such that it cannot exceed a maximum velocity

    Vmax, if that velocity would cause the Compton wavelength to be shorter than a

    Planck length. It turns out that this Vmax is so close to c that most would say it is    

    negligible. But this trivial difference gets rid of the elephant! A calculation for Vmax will

    be made shortly.

7. The Maximum Mass of a Quantum Mechanical Particle:

      Method 1: Using the Lorentz Transform

         1.  λc = [h/(cm0/(1-v2/c2 )0.5)] This is the Compton equation compensated for relativity.

         2.  λc = h/(cM)       Make m0/(1-v2/c2 )0.5 = M

         3.  PL = h/(cM)       The Planck length is the shortest Compton wavelength.

                     At this wavelength the mass M is as great as it can be.

         4.  PL = (hG/c3)0.5      This is the definition of the Planck length

         5.  (hG/c3)0.5 = h/(cM)      Substituting the Planck length into line 3.

         6.  M = (hc/G)0.5              Solving for M.

         7.  M = Pm             This value for M is the Planck mass.


    Thus the Compton wavelength at high velocities becomes the Planck length and the rest    
    mass m
0 becomes the Planck mass.

      Method 212: Using the Einstein-Planck relation E = hf and Einstein’s equation E = mc2

     The diagram (below) is a representation of the mass of an electron (or any particle). This is  

       a visualization of what a particle trapped between mirrors looks like. We can take this diagram to
       the extreme and calculate the maximum possible mass of any quantum particle.

    1. Take the Compton wavelength to its limit of the Planck length = (hG/c3)0.5 .

    2. The energy of this photon is E = hc/λ where λ is now the Planck length.

    3. The energy of this photon is also given by E = mc2 and we have hc/λ = mc2 .

    4. Solving for m we get that m = (hc/G)0.5. This by definition is the Planck mass.

      Again we come to the conclusion that the maximum mass of any quantum particle is
      the Planck mass. The velocity Vmax is attained when a particles Compton wavelength
      has become the Planck length.

8. The Nature of the Lorentz Transform

The Lorentz transform is (1-v2/c2)0.5. When v = Vmax, becomes the exact number needed to:
     1. Make the mass M of all particles the Planck mass
     2. Make the Compton wavelength λ
c of all particles the Planck length.

The Compton wavelength, the Planck length, the Planck mass and the Lorentz contraction all act “in cahoots” to create a maximum velocity Vmax that prevents any particle from getting too close to c.

9. A Calculation for Vmax (the maximum velocity for a quantum mechanical particle)

We can compute the max velocity of a particle Vmax by noting that the particle has a Compton wavelength h/(m0 c) that shrinks to a Planck length when it is at its ultimate speed Vmax . Here is the calculation for the maximum velocity Vmax :

1. PL = h/(m0 c) x (1-Vmax2/c2)0.5 .   We can solve for Vmax as follows:

2. PL2 = h2/(m02 c2) x (1-Vmax2/c2)

3. hG/c3 = h2/(m02 c2) x (1-Vmax2/c2)

4. G = ch/m02 x (1-Vmax2/c2)

5. G/(ch) = 1/m02 x (1-Vmax2/c2), since Pm = (ch/G)0.5 we have,

6. 1/Pm2 = 1/m02 x (1-Vmax2/c2)

7.  m02/Pm2 = 1-Vmax2/c2

8.  Vmax2/c2 = 1- m02/Pm2

9.  Vmax2 = c2(1- m02/Pm2)

     10.  Vmax = c(1- m02/Pm2)0.5    The maximum velocity of a quantum mechanical particle 

     11. We can also calculate the value of the Lorentz transform (1-Vmax2/c2)0.5 at the Vmax

      value. This turns out to be m0/Pm which is a very small number even for the most

      massive of particles like a Buckyball.


10. Vmax for Various Particles Compared to an idealized speed of light:

Speed of Light

3.00000 000xx xxxxx xxxxx xxxxx xxxxx x (10m/s)

Vmax Electron

2.99999 99999 99999 99999 99999 99973 7 (10m/s)

Vmax Proton

2.99999 99999 99999 99999 99117 xxxxx x (10m/s)

Vmax Uranium Atom

2.99999 99999 99999 99498 xxxxx xxxxx x (10m/s)

Vmax Buckyball C60

2.99999 99999 545xx xxxxx xxxxx xxxxx x (10m/s)

The above table was made using the equation Vmax = c(1- m02/Pm2)0.5 . The speed “c” was chosen as an ideal 300,000,000 m/s to make comparisons with the Vmax values easier. This table used the values shown in the spreadsheet below.

The final value for Vmax was obtained using the online calculator at the URL:
http://keisan.casio.com/has10/Free.cgi .
This calculator was needed to get the many digits needed for Vmax .

Note that the less massive a particle the closer its Vmax approaches c. Also note that our current precision on the speed of light is about 1 m/s and that all of the particles listed above need far more precision than that. This difference between Vmax and c may seem minuscule, but this is what limits the ultimate mass of all particles to the Planck mass and not infinity. Yes, NIST made a mistake in making ”c” a constant by fiat!

11. What is the ultimate mass of a Golf Ball:

Let us shift to a classical object instead of a particle. A golf ball with mass 0.0459 kg will do. This golf ball is made up of quantum mechanical particles. That is, particles that can interfere with themselves.

We can make a rough estimate of the limiting mass of the golf ball with velocity. I am sure golfers are very interested in this number. Let us assume that a golf ball is made up of Buckyballs (C60), just to keep things simple. But who knows? It might make a good golf ball. The mass of a Buckyball is 1.19x10-24 kg. This gives 3.86x1022 Buckyballs in a 0.0459 kg golf ball. This number of Buckyballs times the Planck mass (the max mass of each Buckyball) gives the max mass of the golf ball as 8.4x1014 kg. That is a lot of mass but a lot less than infinity and a lot less than the mass of the earth (6x1024 kg).

This expanded analysis removes the elephant not only from the world of particles but also from the world of classical objects. The mass of any object, be it a particle or golf ball, does not approach infinity as its speed approaches the speed of light!

12. The Schwarzschild Radius for Particles:

In the realm of classical physics massive objects (like stars) when they shrink and get denser can approach a radius called the Schwarzschild Radius (r = 2Gm/c2 ) as a limit. Perhaps this kind of phenomena (a consequence of general relativity) is also true of particles. We should check this out.

a. The maximum mass of any particle is the Planck mass (hc/G)0.5

b. If we insert this mass into the formula for the Schwarzschild Radius (r = 2Gm/c2 )

               we get that r = 2GPm/c2

c. Substituting for the Planck mass we get r = 2(G/c2) x (hc/G)0.5

d. Simplifying we get the equation we get r = 2(Gh/c3)0.5.

f. Noting that (Gh/c3)0.5 is the Planck length, we get that r = 2(PL).
          The conclusion is that all particles have a Schwarzschild radius of 2 Planck lengths.

This is interesting! it seems that particles may have a Schwarzschild limit and act as black holes. I believe that John Baez would be an appropriate physicist to give a reasonable answer to this since he has given thought to this issue.13

13. Objections .....This can not be right because:

a. It conflicts with the tried and true theory of special relativity. 14 

        Response: Special relativity does not take into account quantum mechanical limits

        that come into effect as the speed c is approached. This is what is examined in this

        essay. And I believe the insights in this essay build constructively on special relativity.

b. Making the Planck length fundamental is not legitimate physics.

        Response: Sabine Hossenfelder thinks the Planck length is fundamental. 5

           John Baez has the most interesting discussion of length and the Planck

        length in particular. 3

c. Who says that particle mass stops at the Planck mass?

        Response: I do, via the logic contained in this essay. Louie deBroglie’s equation        

        (λ = h/mv) which works very well for an electron, is usually interpreted as valid for any

        mass value. This is a big mistake but not as conspicuous a mistake as having

        electrons capable of approaching infinite mass.

d. How can both an electron and a Buckyball end up with the same ultimate mass value

    of the Planck mass?

        Response: They attain their ultimate mass at different Vmax velocities. The electron

        starts out a lot lighter than a Buckyball, but at its ultimate mass, it is moving closer to the speed

        of light.

e. This is a theory that cannot be tested!

        Response: This is true. At the present time I cannot think of any experiments that can be

        performed to test this theory. Perhaps at some time in the future we will have enough precision

        in measuring speeds that approach c that we can verify this theory directly.

        However, The elephant part of special relativity theory M = m0/(1-v2/c2 )0.5 also can never be

        verified. How can you verify that a mass will approach infinite value?

f. This is nothing but an exercise in numerology.

        Response: I would have to say that the techniques used in this essay could reasonably

        be called “Planck Unit Numerology” (PUN). My suspicion is that in the future Planck Units will be

        accepted as fundamental physical limits and the work done here will be  only a very small

        exploration of a mine that is full of treasures.

14. Summary:

        1. Particles can never be accelerated to “c” because they hit their respective Vmax values first
            and can not be accelerated further. This is because particles are characterized by their
            Compton wavelength and at
Vmax the Compton wavelength has shrunk to the Planck length,
            as short as they can get.
        2. As particles are accelerated to “c” they approach a Schwarzschild radius and what can be
            considered a black hole.

        3. Vmax is different for particles with different rest masses m0 .
Vmax = c(1- m02/Pm2)0.5 .
        4. All rest masses
m0 when they are accelerated to their respective Vmax value:

                a. Have the same Compton wavelength which is the Planck length.

                b. Have the same mass which is the Planck mass.

                c. Have a Lorentz contraction that is equal to m0/Pm

                d. Have a Schwarzschild radius that is two Planck lengths.

        5. Since classical masses (like golf balls) are composed of particles, they like particles

            will not approach infinite mass.



M = m0/(1-v2/c2 )0.5 :  The equation that shows mass increase with velocity. Known as the elephant in
                                   this essay.

(1-v2/c2)0.5 :  Referred to as the Lorentz transform (and Lorentz contraction) in this essay. It is an
                    integral part of special relativity. It’s value goes to zero as v goes to c.

m0 :  The symbol that stands for the rest mass of an object or particle.

v :  The symbol that stands for velocity or speed.

c :  The symbol for the speed of light. Its value is 2.99792458×108 m s−1

λc = h/(m0c) :  The Compton wavelength of a particle when it is at rest. When a particle is moving its
                       Compton wavelength is subject to relativistic shortening and becomes:
                        λc = h/(m0c) x (1-v2/c2)0.5 .

λ = h/mv :  The deBroglie equation that correctly predicts the mass gain of an electron with velocity.
                  The limit for mass in this equation has never been tested beyond the mass of a Buckyball.

h :  The Planck constant. It has a value of 6.626068 × 10-34 m2 kg/s

Buckyball C60 :  The largest molecule that has been shown to exhibit the quantum mechanical
                            wavelike property of interference.

NIST :  National Institute of Standards and Technology, formerly known as the National Bureau of

Planck Length :  PL = (hG/c3)0.5 . It has the value of 1.616 199(97) × 10−35 meter.

Planck Mass :  Pm = (hc/G)0.5 . It has the value of 2.176 51(13) × 10−8 kg.

G :  The gravitational constant. Its value is 6.67384(80)×10−11 m3kg−1s−2

E = hf :  The Planck-Einstein relation and on of the foundations of quantum mechanics.

f :  The frequency of a particle. It is usually tied to the wavelength of the particle via the equation
        c = f(λ) where λ is the wavelength of the particle.

E = mc2 :  Einstein’s relation that links mass and energy.

Vmax = c(1- m02/Pm2)0.5 :  The maximum velocity of a quantum mechanical particle.

Schwarzschild radius (r = 2Gm/c2):  The distance from the center of an object such that, if all the
                                                             mass of the object were compressed within that sphere, the
                                                             escape speed from the surface would equal the speed of light.


1. http://en.wikipedia.org/wiki/Elephant_in_the_room

2. http://en.wikipedia.org/wiki/Mass_in_special_relativity

3. http://math.ucr.edu/home/baez/lengths.html

4. http://physicsworld.com/cws/article/news/1999/oct/15/wave-particle-duality-seen-in-carbon-60-molecules

5. http://backreaction.blogspot.com/2012/01/planck-length-as-minimal-length.html

6. http://www.nist.gov/index.html

7. http://museum.nist.gov/exhibits/ex1/room6.html

8. http://spectrum.ieee.org/consumer-electronics/standards/the-kilogram-reinvented

9. http://en.wikipedia.org/wiki/Planck_units

10. http://en.wikipedia.org/wiki/Compton_wavelength

11. http://www.digitalwavetheory.com/DWT/36_Derivation_of_the_Compton_Wavelength.html

12. http://www.digitalwavetheory.com/DWT/35_Max_Mass_of_a_Quantum_Particle.html

13. http://math.ucr.edu/home/baez/planck/node2.html

14. http://www.nature.com/nature/journal/v450/n7171/full/450801a.html

All the web references above have been verified as of 8/04/2012 by Don Limuti

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