[General] Connection between Radii

[John Macken]

Hello All,


 


This second post shows that there is a previously unrecognized relationship between a particle’s reduced Compton wavelength λc and the particle’s Schwarzschild radius.  This post will show the reason why this relationship exists.  In my “Foundation” paper and “Aether” papers (attached to my previous post) I state that spacetime has a single field rather than the 17 overlapping fields of the standard model.  The “spacetime field” has multiple resonances which correspond to the Compton frequencies of the various fundamental particles of the standard model.  The spacetime field is a sea of dipole waves in spacetime with dimensionless strain amplitude:  


 


    As = Lp/λ = Tpω  


 


These waves are all frequencies up to Planck angular frequency ωp but there are also various resonant frequencies which we know as virtual particles. In the Foundation paper I explain that I view gravity as a nonlinear effect which scales with wave amplitude squared (to a first approximation).  Equations 15 to 21 of the Foundation paper support this prediction.  For example, the only difference between the gravitational force Eq. 15 and the electrostatic force Eq. 16 is that the strain amplitude in Eq. 15 is As2 and in Eq. 16 this is As (not squared).  Other equations also show that gravity is a nonlinear effect which scales with amplitude squared As2.


 


Now I am ready to give my new insight.  I have discovered that there is a fundamental relationship between a particle’s Compton radius λc and its Schwarzschild radius Rs.  Most important, I have discovered the reason for this relationship and it perfectly supports my model of the universe.  In my model of a fundamental particle (my rotar model) is a dipole wave in spacetime rotation at the speed of light at its Compton angular frequency ωc. It is one Compton wavelength in radius which means that its mathematical radius is its reduced Compton wavelength λc = ħ/mc = c/ωc (single loop).  This is the natural unit of length for a fundamental particle and is the reason that force equations become very simple when distance is expressed as the number (N) of reduced Compton wavelengths rather than meters (N = r/λc).  In order to mathematically analyze the particle characteristics, it is necessary to also specify the wave amplitude and the impedance of spacetime.  The particle’s strain amplitude is As = Lp/λc = Tpωc and the impedance of spacetime is: Zs = c3/G ≈ 4x1035 kg/s. This combination of component parts has been analyzed in the Foundation paper and the result is that the particle model has internal energy Ei = ħωc which is correct.  


 


The Schwarzschild radius of this rotating wave particle model is: Rs = Gm/c2  rather than  rs = 2Gm/c2 .  The reason for this difference is because the single fundamental particle is maximally rotating at the speed of light.  The difference is a factor of 2 compared to the Schwarzschild radius of a non-rotating body made of multiple particles.  I make a distinction by using the symbol Rs rather than rs.  


 


The Foundation paper shows that there is a square relationship between the electrostatic force and the gravitational force when forces between single particles are expressed in their fundamental units.  That paper also shows that there is a symmetrical relationship between the gravitational force, the electrostatic force and Planck force.  Now I have determined that there is also a fundamental relationship between a fundamental particle’s radius λc = ħ/mc and its Schwarzschild radius Rs = Gm/c2.  Again, this relationship requires that these radii be expressed in the natural units which are dimensionless Planck units (designated with an underline) 


  


λcRs = 1      

where:  λs = λs/Lp    and     Rs = Rs/Lp          Planck length =  Lp = (ħG/c3)1/2

In words, a fundamental particle’s Compton radius times its Schwarzschild radius equals 1 when both radii are expressed in dimensionless Planck units. Another way of expressing this is:

  λc/Lp = Lp/Rs

There is a type of symmetry between a particle’s Compton radius  λc, a particle’s Schwarzschild radius and Planck length.  For example, imagine a log scale of length. On this scale we designate a particle’s Compton radius λc and its Schwarzschild radius Rs.  Planck length is exactly half way between these two designation points on this log scale. There is another important connection between the radii and the forces.  Suppose that we have two of the same particles, both with elementary charge e.  If they are separated by an arbitrary distance, they will experience an electrostatic force (Fe) and a gravitational force (Fg).  The ratio of these two forces is constant (independent of separation distance).  This ratio exactly equals the ratio of radii Rs/λc if we compensate for the electrostatic coupling constant by incorporating the inverse of the fine structure constant α-1 = 137.036 

  

For example, for an electron, both ratios equal 1.75 x 10-45. 

However, I was looking for the reason that there should be a relationship between λc and Rs.  These are both radii where Rs is the gravitational radius and λc is a particle’s radius (single loop).  In my model, a black hole happens when the strain of spacetime reaches 100% for a particular wavelength or radius.  Another way of saying this is a black hole happens when As = 1 for a particular wavelength.  To explain the rest of the concept I am going to use an electron as an example.  An electron has strain amplitude As = ħ/mc = m/mp = ωc/ωp = Lp/λc ≈ 4.18 x 10-23 (dimensionless).  The standard model also recognizes the importance of this number because the standard model contains many dimensionless numbers and this is the number associated with electrons.  In my model, the spatial distortion produced by the rotating dipole wave is As = Lp/λc.  Gravitational effects scale with As2 because the spacetime field is finite and has boundary conditions which means that it is a nonlinear medium for wave propagation.  To a first approximation the nonlinear effect scales with amplitude squared.  There are higher order terms which are being ignored.  Therefore, the oscillating gravitational amplitude is As2 ≈ 1.75x10-45 for an electron.  In order to reach the condition which would produce an gravitational amplitude of As2 = 1 we would have to reduce the rotar radius of the electron by a factor of As2. A mechanical reduction of the radius would both increase the amplitude As and increase the Compton frequency ωc. In the mathematical calculation of the Schwarzschild radius, there is no mechanical reduction of radius.  Therefore there would be no increase in ωc. Therefore it is necessary for the calculation to increase the amplitude by As2 in order to calculate the Schwarzschild radius with no change in frequency.  This leads to the new insight which perfectly supports this model.  Here is the equation: 


    Rs = λcAs2   


    For an electron: Rs = 6.76x10-58 m,     λc = 3.86x10-13 m    and As2 = 1.75x10-45  


 


I have generated a specific quantifiable models of an electron, a photon, the energy density of spacetime, the curvature of spacetime, the gravitational force, the electrostatic force between particles, an electric field and a charged particle.  I have found it particularly difficult to get anyone in this group to discuss any of these points.  If you disagree with anything in my model, I would like you to point out the logical flaw.  I would be happy to discuss and if necessary to debate any point in my model.  It would actually be very helpful to me if you can point out a logical flaw. 

 

John M. 

 

 

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